boosting the margin a new explanation for the effectiveness of voting methods one of the surprising recurring phenomena observed in experiments with boosting is that the test error of the generated classifier usually does not increase as its size becomes very large and often is observed to decrease even after the training error reaches zero in this paper we show that this phenomenon is related to the distribution of margins of the training examples with respect to the generated voting classification rule where the margin of an example is simply the difference between the number of correct votes and the maximum number of votes received by any incorrect label we show that techniques used in the analysis of vapnik s support vector classifiers and of neural networks with small weights can be applied to voting methods to relate the margin distribution to the test error we also show theoretically and experimentally that boosting is especially effective at increasing the margins of the training examples finally we compare our explanation to those based on the bias variance decomposition

time dependent inverse scattering from gradient type interfaces using an exact solution in the context of one dimensional linear wave propagation a gradient type interface is a point at which the velocity profile of a scattering medium suffers a jump in first derivative in this paper a time domain approach to scattering from such an interface leads to an integrodifferential equation involving the kernel of the reflection operator impulse response which can be solved exactly if the velocity profile is piecewise parabolic this special case provides the basis for an approximate numerical scheme to solve the inverse scattering problem recover the velocity profile from reflection data for a general velocity profile the algorithm is fast and affords good results if the velocity profile is nearly parabolic or the interest is in short time only a detailed error estimate for the approximation is provided

out of sample extrapolation of learned manifolds we investigate the problem of extrapolating the embedding of a manifold learned from finite samples to novel out of sample data we concentrate on the manifold learning method called maximum variance unfolding mvu for which the extrapolation problem is still largely unsolved taking the perspective of mvu learning being equivalent to kernel pca our problem reduces to extending a kernel matrix generated from an unknown kernel function to novel points leveraging on previous developments we propose a novel solution which involves approximating the kernel eigenfunction using gaussian basis functions we also show how the width of the gaussian can be tuned to achieve extrapolation experimental results which demonstrate the effectiveness of the proposed approach are also included

factorization of the dissipative wave equation and inverse scattering a decomposition of the solution of the dissipative wave equation into incoming and outgoing components across a smooth surface in a homogeneous region is presented the proof of the decomposition is given only for the plane surface this is then applied to the factorization of the dissipative wave equation into incoming and outgoing components in a planar stratified medium the ricatti integral differential equation for the reflection operator that relates the two components is obtained it is shown how the zeroth and second moments with respect to the transverse variable in a planar stratified medium of the reflection kernel can be used in the inverse problem to recover the velocity and dissipation coefficient from knowledge of the scattered field

manifold regularization a geometric framework for learning from labeled and unlabeled examples we propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution we focus on a semi supervised framework that incorporates labeled and unlabeled data in a general purpose learner some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases we use properties of reproducing kernel hilbert spaces to prove new representer theorems that provide theoretical basis for the algorithms as a result in contrast to purely graph based approaches we obtain a natural out of sample extension to novel examples and so are able to handle both transductive and truly semi supervised settings we present experimental evidence suggesting that our semi supervised algorithms are able to use unlabeled data effectively finally we have a brief discussion of unsupervised and fully supervised learning within our general framework

general tensor discriminant analysis and gabor features for gait recognition traditional image representations are not suited to conventional classification methods such as the linear discriminant analysis lda because of the undersample problem usp the dimensionality of the feature space is much higher than the number of training samples motivated by the successes of the two dimensional lda 2dlda for face recognition we develop a general tensor discriminant analysis gtda as a preprocessing step for lda the benefits of gtda compared with existing preprocessing methods such as the principal components analysis pca and 2dlda include the following 1 the usp is reduced in subsequent classification by for example lda 2 the discriminative information in the training tensors is preserved and 3 gtda provides stable recognition rates because the alternating projection optimization algorithm to obtain a solution of gtda converges whereas that of 2dlda does not we use human gait recognition to validate the proposed gtda the averaged gait images are utilized for gait representation given the popularity of gabor function based image decompositions for image understanding and object recognition we develop three different gabor function based image representations 1 gabord is the sum of gabor filter responses over directions 2 gabors is the sum of gabor filter responses over scales and 3 gaborsd is the sum of gabor filter responses over scales and directions the gabord gabors and gaborsd representations are applied to the problem of recognizing people from their averaged gait images a large number of experiments were carried out to evaluate the effectiveness recognition rate of gait recognition based on first obtaining a gabor gabord gabors or gaborsd image representation then using gdta to extract features and finally using lda for classification the proposed methods achieved good performance for gait recognition based on image sequences from the university of south florida usf humanid database experimental comparisons are made with nine state of the art classification methods in gait recognition

wave splitting and the reflection operator for the wave equation in r3 the problem of wave splitting in a nonhomogeneous medium in r3 is considered previous results for wave splitting in a planar stratified medium can be generalized to a general nonhomogeneous medium with sufficiently smooth velocity the wave equation is factorized into an up and down going wave system using certain integral and integral differential operators the equation for the reflection operator which relates the up going wave to a down going wave is then obtained and certain properties of the reflection operator are deduced

simultaneous reconstruction of material and transient source parameters using the invariant imbedding method this paper extends the time domain wave splitting and invariant imbedding method to an inhomogeneous wave equation with a source term uxx utt a x ux 2d x i t the direct scattering and inverse source problems of this equation are studied operators j that map the source function into the scattered waves at the edges of the slab are defined a system of coupled nonlinear integrodifferential equations for these scattering operator kernels is obtained the direct scattering problem is to obtain the scattering operator kernels j and r when parameters a and d are given the inverse problem is to simultaneously reconstruct a x and d x from the scattering operator kernels r 0 t 0 t 2 and j 0 t 0 t 1 both numerical inversion algorithms and the small time approximate reconstruction method are presented a green s function technique is used to derive green s operator kernel equations for the calculation of the internal field it provides an alternative effective and fast way to compute the scatterin

an alternative perspective on adaptive independent component analysis algorithms this article develops an extended independent component analysis algorithm for mixtures of arbitrary subgaussian and supergaussian sources the gaussian mixture model of pearson is employed in deriving a closedform generic score function for strictly subgaussian sources this is combined with the score function for a unimodal supergaussian density to provide a computationally simple yet powerful algorithm for performing independent component analysis on arbitrary mixtures of nongaussian sources

the calabi yau conjectures for embedded surfaces in this talk i will discuss the proof of the calabi yau conjectures for embedded surfaces this is joint work with bill minicozzi cm9 the calabi yau conjectures about surfaces date back to the 1960s much work has been done on them over the past four decades in particular examples of jorge xavier from 1980 and nadirashvili from 1996 showed that the immersed versions were false we will show here that for embedded surfaces i e injective immersions they are in fact true their original form was given in 1965 in ca where e calabi made the following two conjectures about minimal surfaces see also s s chern page 212 of ch conjecture 1 prove that a complete minimal hypersurface in r must be unbounded calabi continued it is known that there are no compact minimal submanifolds of r or of any simply connected complete riemannian manifold with sectional curvature 0 a more ambitious conjecture is conjecture 2 a complete minimal hypersurface in r has an unbounded projection in every n 2 dimensional flat subspace

locally linear discriminant analysis for multimodally distributed classes for face recognition with a single model image we present a novel method of nonlinear discriminant analysis involving a set of locally linear transformations called locally linear discriminant analysis llda the underlying idea is that global nonlinear data structures are locally linear and local structures can be linearly aligned input vectors are projected into each local feature space by linear transformations found to yield locally linearly transformed classes that maximize the between class covariance while minimizing the within class covariance in face recognition linear discriminant analysis lida has been widely adopted owing to its efficiency but it does not capture nonlinear manifolds of faces which exhibit pose variations conventional nonlinear classification methods based on kernels such as generalized discriminant analysis gda and support vector machine svm have been developed to overcome the shortcomings of the linear method but they have the drawback of high computational cost of classification and overfitting our method is for multiclass nonlinear discrimination and it is computationally highly efficient as compared to gda the method does not suffer from overfitting by virtue of the linear base structure of the solution a novel gradient based learning algorithm is proposed for finding the optimal set of local linear bases the optimization does not exhibit a local maxima problem the transformation functions facilitate robust face recognition in a low dimensional subspace under pose variations using a single model image the classification results are given for both synthetic and real face data

the existence of least area surfaces in 3 manifolds this paper presents a new and unified approach to the existence theorems for least area surfaces in 3 manifolds introduction a surface f smoothly embedded or immersed in a riemannian manifold m is minimal if it has mean curvature zero at all points it is a least area surface in a class of surfaces if it has finite area which realizes the infimum of all possible areas for surfaces in this class the connection between these two ideas is that a surface which is of least area in a reasonable class of surfaces must be minimal the converse is false minimal surfaces are in general only critical points for the area function there are close analogies between these two concepts and the theory of geodesics in a riemannian manifold minimal surfaces correspond to geodesics and least area surfaces correspond to geodesic arcs or closed geodesics which have shortest length in some class of paths any geodesic a in a riemannian manifold m has the property that it is locally shortest i e if p and q are nearby points on a then the subarc of a which joins p and q is the shortest path in m from p to q it can also be proved that minimal surfaces are locally of least area but the proof is difficult and involves substantial knowledge of the theory of partial differential equations there are now a large number of theorems asserting the existence of surfaces of least area in various classes surfaces of this type have become an important tool in 3 dimensional topology in this paper we present a new approach to the proofs of these existence theorems this yields a simplified and unified method for the proof of the existence of minimal surfaces in 3 dimensional riemannian manifolds in 1930 douglas do and rado ra independently showed that a simple closed curve in ri which bounds a disk of finite area bounds a disk of least possible area this result was extended by morrey moi in 1948 to a general class of riemannian manifolds the homogeneously regular manifolds a class which includes all closed manifolds work of osserman 0 and gulliver g later showed that least area disks in 3 manifolds were immersed in their interiors more recently there has been a series of new existence results for closed surfaces of least area in manifolds of any dimension it follows from work of sacks and uhlenbeck s ui that if m is closed and 7r2 m is nonzero then there is an essential map of the 2 sphere into m which has least area among all essential maps sacks and uhlenbeck s uii and schoen and yau s y independently showed that if f f m is a map of a closed orientable surface f not the 2 sphere into a closed riemannian manifold m such received by the editors june 23 1986 1980 mathematics subject classification 1985 revision primary 53a10 secondary 57m35 the first author was supported by nsf postdoctoral fellowship dms84 14097 1988 american mathematical society 0002 9947 88 1 00 25 per page

inverse scattering method for one dimensional inhomogeneous layered media an inverse scattering method to reconstruct simultaneously the permittivity profile and the conductivity profile of one dimensional inhomogeneons medium which makes use of the transverse electric te wave and or transverse magnetic tm wave is proposed the medium is illuminated by the te and or tm plane wave at oblique incidence and the data are taken as the reflection coefficients for a set of discrete frequencies and or a finite number of incident angles furthermore the reflection coefficient data contain the gaussian noise the nonlinear integral equation relating the unknown constitutive parameter of the medium to the reflection coefficient for te wave and or tm wave is solved by the newton iteration method the inverse operator in the newton method is determined by the regularization method it is demonstrated in terms of the numerical examples that this method utilizing both polarizations and the incident angle of the incident plane wave is very effective even if the reflection coefficient contains the practical measurement error or the phase of the reflection coefficient is unknown moreover the relationships between the errors of reconstructed profile and the measured reflection coefficient are also discussed

alignment of overlapping locally scaled patches for multidimensional scaling and dimensionality reduction data observations that lie on a manifold can be approximated by a collection of overlapping local patches the alignment of which in a low dimensional euclidean space provides an embedding of the data this paper describes an embedding method using classical multidimensional scaling as a local model based on the fact that a manifold locally resembles a euclidean space a set of overlapping neighborhoods are chosen by a greedy approximation algorithm of minimum set cover local patches derived from the set of overlapping neighborhoods by classical multidimensional scaling are aligned in order to minimize a residual measure which has a quadratic form of the resulting global coordinates and can be minimized analytically by solving an eigenvalue problem this method requires only distances within each neighborhood and provides locally isometric embedding results the size of the eigenvalue problem scales with the number of overlapping neighborhoods rather than the number of data points experiments on both synthetic and real world data sets demonstrate the effectiveness of this method extensions and variations of the method are discussed

a general framework for dimensionality reducing data visualization mapping in recent years a wealth of dimension reduction techniques for data visualization and preprocessing has been established nonparametric methods require additional effort for out of sample extensions because they provide only a mapping of a given finite set of points in this letter we propose a general view on nonparametric dimension reduction based on the concept of cost functions and properties of the data based on this general principle we transfer nonparametric dimension reduction to explicit mappings of the data manifold such that direct out of sample extensions become possible furthermore this concept offers the possibility of investigating the generalization ability of data visualization to new data points we demonstrate the approach based on a simple global linear mapping as well as prototype based local linear mappings in addition we can bias the functional form according to given auxiliary information this leads to explicit supervised visualization mappings with discriminative properties comparable to state of the art approaches

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

factorization of the wave equation in higher dimensions the factorization of the wave equation into a coupled system involving up and down going wave components is obtained for the case where the field quantities are multivariate functions of spatial variables but the velocity c is a function of the z variable only the form of the reflection operator is derived and the quadratic differential integral equation satisfied by its kernel is obtained

non properly embedded h planes in mathbb h 2 times mathbb r for any h in 0 frac 1 2 we construct complete non proper stable simply connected surfaces embedded in mathbb h 2 times mathbb r with constant mean curvature h

existence uniqueness and causality theorems for wave propagation in stratified temporally dispersive complex media a mixed initial boundary value problem for a nonlocal hyperbolic equation is analyzed with respect to unique solubility and causality the regularity of the step response and impulse response the green functions is investigated and a wave front theorem is proved the problem arises e g at time varying electromagnetic plane wave excitation of stratified temporally dispersive bi isotropic or anisotropic slabs concluding the problem is uniquely solvable strict causality holds and a well defined wave front speed exists this speed is independent of dispersion and excitation and depends on the nondispersive properties of the medium only

a hyperspectral image classification framework and its application a new descriptor called lrp is proposed to describe subtle texture for 2d image texture enhancement is built to extend lrp to represent hsi a supervised manifold learning method called dlpp is proposed to reduce dimension the proposed framework outperforms other methods with experiments classification is a critical step to make full use of the hyperspectral data the most current approaches perform well for analyzing the macro texture but they often fail to deal with the micro texture thus this study proposes a general framework for the material with micro texture based on hyperspectral image hsi technique in this framework local response pattern lrp is firstly proposed to describe 2d image texture to preserve more structural information and keep less sensitive to image conditions then lrp is extended to represent hsi with texture enhancement te by considering opponent relationships between pairs of bands after that discriminated locality preserving projection dlpp is proposed to reduce data dimension in a linearizing nonlinear manifold way finally experiments on the hyperspectral images of fresh and frozen thawed fish fillets are conducted the results demonstrate that the proposed framework is efficient in terms of both recognition rates and robustness

an electromagnetic inverse problem for dispersive media the dispersion of transient electromagnetic waves in a homogeneous medium can be characterized by expressing either the complex permittivity as a function of frequency or the susceptibility kernel as a function of time in this paper a time domain technique is used to derive a nonlinear integrodifferential equation which relates the susceptibility kernel for a one dimensional homogeneous slab to the reflection operator for the medium thus the susceptibility kernel which is a function of time can be determined from reflection data a numerical implementation of this technique is shown the more general case of a medium consisting of a stack of homogeneous dispersive layers is also addressed

discriminant adaptive nearest neighbor classification nearest neighbour classification expects the class conditional probabilities to be locally constant and suffers from bias in high dimensions we propose a locally adaptive form of nearest neighbour classification to try to ameliorate this curse of dimensionality we use a local linear discriminant analysis to estimate an effective metric for computing neighbourhoods we determine the local decision boundaries from centroid information and then shrink neighbourhoods in directions orthogonal to these local decision boundaries and elongate them parallel to the boundaries thereafter any neighbourhood based classifier can be employed using the modified neighbourhoods the posterior probabilities tend to be more homogeneous in the modified neighbourhoods we also propose a method for global dimension reduction that combines local dimension information in a number of examples the methods demonstrate the potential for substantial improvements over nearest neighbour classification

transient wave propagation in reciprocal bi isotropic media at oblique incidence in this paper a new wave splitting is suggested that simplifies the analysis of wave propagation in reciprocal bi isotropic media two different methods to solve the scattering problem are analyzed the invariant imbedding and the green function approach the medium is modeled by constitutive relations in the time domain time convolutions and the slab is assumed to be inhomogeneous with respect to the depth it is shown that the cross polarized contribution to the reflected field at normal incidence is zero for the general reciprocal inhomogeneous slab moreover the rotation and the attenuation of the wave front are calculated explicitly in the general inhomogeneous slab case special attention is paid to normal incidence and to the homogeneous semi infinite medium

a numerical solution to the zakharov shabat inverse scattering problem a novel numerical method to solve the inverse scattering problem associated with the zakharov shabat coupled mode equations is developed in this problem the coupling coefficient for two propagating modes is determined from reflection data both codirectional and contradirectional coupling can be accommodated as well as the presence of bound states analytical results based on an independent analytical method are derived in order to demonstrate the accuracy of the proposed method which is applicable to a broad class of reflection data

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

embeddedness of the solutions to the h plateau problem abstract we generalize meeks and yau s embeddedness result on the solutions of the plateau problem to constant mean curvature disks we show that any minimizing h disk in an h 0 convex domain is embedded for any h 0 h 0 in particular for the unit ball b in r 3 this implies that for any h 0 1 any jordan curve in b bounds an embedded h disk in b

on almost fuchsian manifolds almost fuchsian manifold is a class of complete hyperbolic threemanifolds a such three manifold is a quasi fuchsian manifold that contains a closed incompressible minimal surface with principal curvatures everywhere in the range of 1 1 in such a manifold the minimal surface is unique and embedded hence one can parametrize these hyperbolic three manifolds by their minimal surfaces in this paper we obtain estimates on several geometric and analytical quantities of an almost fuchsian manifold m in terms of the data on the minimal surface in particular we obtain an upper bound for the hyperbolic volume of the convex core of m and an upper bound on the hausdorff dimension of the limit set associated to m we also constructed a quasi fuchsian manifold which admits more than one minimal surface and it does not admit a foliation of closed surfaces of constant mean curvature

the existence of embedded minimal surfaces and the problem of uniqueness it was in the early nineteen thirties that douglas and radd solved the plateau problem for a rectifiable jordan curve in euclidean space the minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk in 1948 morrey generalized the theorem of douglas and radd to minimal surfaces in a homogeneously regular riemannian manifold in both cases the solutions are defective as they may possess branch points it was not until 1968 that osserman 24 was able to prove that interior branch points do not exist on the douglas radd morrey solution osserman s theorem in this full generality was proved by alt 2 3 and gulliver 8 if the jordan curve is real analytic gulliver and lesley 9 showed that the solution surface is free of branch points even on its boundary hence at least in this case we know that the douglas radd morrey solution is an immersion and represents a classical surface however the douglas radd morrey solution may in general have selfintersections it was then a classical problem to find conditions on the jordan curve to guarantee that the solution is an embedded disk in 1932 radd was the first to produce a general condition of this type he proved that if the curve can be projected onto a convex curve in the plane then it bounds a unique embedded minimal disk which is a graph over this plane it was not until the seventies that there was further progress on the problem of embeddedness osserman had conjectured that if a jordan curve is extreme that is lies on the boundary of its convex hull then the douglas rad6 solution is embedded gulliver and spruck 10 proved this conjecture under the additional assumption that the absolute total curvature of the jordan curve is less than 4 z right after this work tomi and tromba 32 showed that an extreme curve must bound an embedded minimal disk which need not be stable in 1976 almgren and simon 1 established the existence for such a curve of an embedded stable minimal disk which need not be a douglas radd solution finally in the same year the authors 18 proved that any douglas rad6 solution must be embedded

general curvature estimates for stable h surfaces in 3 manifolds applications we obtain an estimate for the norm of the second fundamental form of stable h surfaces in riemannian 3 manifolds with bounded sectional curvature our estimate depends on the distance to the boundary of the surface and on the bound on the sectional curvature but not on the manifold itself we give some applications in particular we obtain an interior gradient estimate for h sections in killing submersions

hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity in this paper we study the problem of finding smooth complete hypersurfaces of constant mean curvature in hyperbolic space with a prescribed asymptotic boundary at infinity our main results are proved by deriving a priori global gradient estimates based on the classical maximum principle for elliptic equations

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

generic uniqueness of area minimizing disks for extreme curves we show that for a generic nullhomotopic simple closed curve gamma in the boundary of a compact orientable mean convex 3 manifold m with h 2 m bf z 0 there is a unique area minimizing disk d embedded in m with partial d gamma we also show that the same is true for nullhomologous curves in the absolutely area minimizing surface case

classification of gene expression data the manifold based metric learning way classification of microarray gene expression data can potentially help medical diagnosis and becomes an important topic in bioinformatics however microarray data sets are usually of small sample size relative to an overwhelming number of genes this makes the classification problem fairly challenging instance based learning ibl algorithms such as nearest neighbor k nn are usually the baseline algorithm due to their simplicity however practices show that k nn performs not very well in this field this paper introduces manifold based metric learning to improve the performance of ibl methods a novel metric learning algorithm is proposed by utilizing both local manifold structural information and local discriminant information in addition a random subspace extension is also presented we apply the proposed algorithm to the gene classification problem in three ways one in the original feature space another in the reduced feature space and the third via the random subspace extension statistical evaluation shows that the proposed algorithm can achieve promising results and gain significant performance improvement over traditional ibl algorithms

dimensionality reduction an interpretation from manifold regularization perspective in this paper we propose to unify various dimensionality reduction algorithms by interpreting the manifold regularization mr framework in a new way although the mr framework was originally proposed for learning we utilize it to give a unified treatment for many dimensionality reduction algorithms from linear to nonlinear supervised to unsupervised and single class to multi class approaches in addition the framework can provide a general platform to design new dimensionality reduction algorithms the framework is expressed in the form of a regularized fitting problem in a reproducing kernel hilbert space it consists of one error part and two regularization terms the complexity term and the smoothness term the error part measures the difference between the estimated low dimensional data distribution and the true high dimensional data distribution or the difference between the estimated and targeted low dimensional representations of data the complexity term is a measurement of the complexity of the feature mapping for dimensionality reduction and the smoothness term reflects the intrinsic structure of data based on the framework we propose a manifold regularized kernel least squares mr kls method which can efficiently learn an explicit feature mapping in the semi supervised sense experiments show that our approach is effective for out of sample extrapolation c 2014 elsevier inc all rights reserved

discrete inverse method for viscoelastic medium with complete data the discrete inverse scattering problem for viscoelastic medium is studied in this paper it is assumed that the relaxation modulus varies only with time t the object of this paper is to develop a method to reconstruct the relaxation modulus with less measurement data than before the propagation operators of the viscoelastic medium are defined first and the imbedding equations governing the behavior of the propagation operators are derived with the invariant imbedding techniques using the finite difference method these equations can be discretized to obtain a system of linear algebraic equations about the propagation operators and the material modulus for the inverse scattering problem it is assumed that the reflection data obtained from the scattering experiments are only available on one side of the medium and for one round trip through the viscoelastic slab to reconstruct the unknown relaxation modulus an inversion procedure is developed using this set of data that are complete in the sense that they can be extended to arbitrary time t and the other scattering and propagation operators can also be determined by the inversion procedure described in this paper the inversion algorithm is implemented numerically on several examples at the end of the paper it can be seen that the obtained curves of the material modulus coincide with the original relaxation modulus very well

passage method for nonlinear dimensionality reduction of data on multi cluster manifolds nonlinear dimensionality reduction of data lying on multi cluster manifolds is a crucial issue in manifold learning research an effective method called the passage method is proposed in this paper to alleviate the disconnectivity short circuit and roughness problems ordinarily encountered by the existing methods the specific characteristic of the proposed method is that it constructs a globally connected neighborhood graph superimposed on the data set through technically building the smooth passages between separate clusters instead of supplementing some rough inter cluster connections like some existing methods the neighborhood graph so constructed is naturally configured as a smooth manifold and hence complies with the effectiveness condition underlying manifold learning this theoretical argument is supported by a series of experiments performed on the synthetic and real data sets residing on multi cluster manifolds

inverse scattering for inhomogeneous viscoelastic media in this paper the inverse scattering problems for the full inhomogeneous viscoelastic medium are studied via the invariant imbedding technique special attention is paid to the propagation operators of the viscoelastic medium and the imbedding equations for these operators are derived for the inverse scattering problems it is shown that the reflection data can be extended from one round trip through the iscoelastic slab to arbitrary time with the help of the propagation operators hence the reconstruction of the relaxation modulus is sufficient to be considered only in one round trip it is also shown that only one side measurement reflection data are not sufficient to reconstruct the relaxation modulus and the density of the medium simultaneously the corresponding numerical examples are presented for the case that the relaxation modulus of the medium is modeled by two independent functions an iterative inversion procedure is proposed to recover the relaxation modulus and the density simultaneously with the input two side normally reflection data

kernel dimension reduction in regression we present a new methodology for sufficient dimension reduction sdr our methodology derives directly from the formulation of sdr in terms of the conditional independence of the covariate x from the response y given the projection of x on the central subspace cf j amer statist assoc 86 1991 316 342 and regression graphics 1998 wiley we show that this conditional independence assertion can be characterized in terms of conditional covariance operators on reproducing kernel hilbert spaces and we show how this characterization leads to an m estimator for the central subspace the resulting estimator is shown to be consistent under weak conditions in particular we do not have to impose linearity or ellipticity conditions of the kinds that are generally invoked for sdr methods we also present empirical results showing that the new methodology is competitive in practice

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

efficient isometric multi manifold learning based on the self organizing method geodesic distance as an essential measurement for data similarity has been successfully used in manifold learning however many geodesic based isometric manifold learning algorithms such as the isometric feature mapping isomap and geonlm fail to work on data that distribute on clusters or multiple manifolds this limits their applications because practical data sets generally distribute on multiple manifolds in this paper we propose a new isometric multi manifold learning method called multi manifold proximity embedding mpe which can be efficiently optimized using the gradient descent method or the self organizing method compared with the previous methods the proposed method has two steps which can isometrically learn data distributed on several manifolds and is more accurate in preserving both the intra manifold and the inter manifold geodesic distances the effectiveness of the proposed method in recovering the nonlinear data structure and clustering is demonstrated through experiments on both synthetically and real data sets

attributes reduction using fuzzy rough sets fuzzy rough sets are the generalization of traditional rough sets to deal with both fuzziness and vagueness in data the existing researches on fuzzy rough sets are mainly concentrated on the construction of approximation operators less effort has been put on the attributes reduction of databases with fuzzy rough sets this paper mainly focuses on the attributes reduction with fuzzy rough sets after analyzing the previous works on attributes reduction with fuzzy rough sets we introduce formal concepts of attributes reduction with fuzzy rough sets and completely study the structure of attributes reduction an algorithm using discernibility matrix to compute all the attributes reductions is developed based on these lines of thought we set up a solid mathematical foundation for attributes reduction with fuzzy rough sets the experimental results show that the idea in this paper is feasible and valid

transient waves in nonstationary media this paper treats propagation of transient waves in nonstationary media which has many applications in for example electromagnetics and acoustics the underlying hyperbolic equation is a general homogeneous linear first order 2 2 system of equations the coefficients in this system depend on one spatial coordinate and time furthermore memory effects are modeled by integral kernels which in addition to the spatial dependence are functions of two different time coordinates these integrals generalize the convolution integrals frequently used as a model for memory effects in the medium specifically the scattering problem for this system of equations is addressed this problem is solved by a generalization of the wave splitting concept originally developed for wave propagation in media which are invariant under time translations and by an imbedding or a green s functions technique more explicitly the imbedding equation for the reflection kernel and the green s functions propagator kernels equations are derived special attention is paid to the problem of nonstationary characteristics a few numerical examples illustrate this problem

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

reconstruction of source and medium parameters via wave splitting and green function equations this paper applies the time domain wave splitting and the green function concepts to study two inverse problems involving an internal transient source embedded in a nondispersive inhomogeneous medium these two inverse problems are 1 given the physical one way travel time green operator kernel bar g 0 t and the source space distribution function reconstruct the wave speed profile c z of the medium 2 given the same kernel bar g 0 t and the medium wave speed function recover the source space distribution profile p z both numerical inversion algorithms and approximate reconstruction formulas are presented several computational examples are given

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

direct and inverse scattering in the time domain via invariant imbedding equations this paper examines the implementation of a new approach to direct and inverse scattering problems in the time domain which is applied here to the case of a one dimensional lossless medium the method is based on an integro differential equation which is satisfied by the reflection kernel and has been derived elsewhere using invariant imbedding techniques an example of an exact solution to this equation is given numerical schemes for solving the direct and inverse problems are derived by discretizing this equation the behavior of the resulting algorithms is then tested on several examples the results suggest that the inversion method is stable in the presence of additive noise

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

the random subspace method for constructing decision forests much of previous attention on decision trees focuses on the splitting criteria and optimization of tree sizes the dilemma between overfitting and achieving maximum accuracy is seldom resolved a method to construct a decision tree based classifier is proposed that maintains highest accuracy on training data and improves on generalization accuracy as it grows in complexity the classifier consists of multiple trees constructed systematically by pseudorandomly selecting subsets of components of the feature vector that is trees constructed in randomly chosen subspaces the subspace method is compared to single tree classifiers and other forest construction methods by experiments on publicly available datasets where the method s superiority is demonstrated we also discuss independence between trees in a forest and relate that to the combined classification accuracy

complete minimal varieties in hyperbolic space it is a basic problem in the subject of minimal varieties to determine the class of complete minimal and area minimizing varieties m k in a given riemannian manifold n besides being of their own intrinsic interest the cases r of euclidean space and mn sn of the sphere are fundamental in understanding the local structure of minimal varieties due to the existence of tangent cones to general minimal varieties there has been much study of the complete areaminimizing varieties in these cases in particular relating to the bernstein conjecture and minimizing cones e g l2 f1 nevertheless many deep problems remain unsolved this paper is centered around the question of existence of complete areaminimizing varieties in hyperbolic space h here by way of contrast to the situation in ir n or s we find a comparatively complete solution to the problem of existence in fact the main result of this paper exhibits a wealth of complete area minimizing varieties in h in all dimensions and codimensions more precisely we have