mixture subclass discriminant analysis link to restricted gaussian model and other generalizations in this paper a theoretical link between mixture subclass discriminant analysis msda and a restricted gaussian model is first presented then two further discriminant analysis da methods i e fractional step msda fsmsda and kernel msda kmsda are proposed linking msda to an appropriate gaussian model allows the derivation of a new da method under the expectation maximization em framework em msda which simultaneously derives the discriminant subspace and the maximum likelihood estimates the two other proposed methods generalize msda in order to solve problems inherited from conventional da fsmsda solves the subclass separation problem that is the situation in which the dimensionality of the discriminant subspace is strictly smaller than the rank of the inter between subclass scatter matrix this is done by an appropriate weighting scheme and the utilization of an iterative algorithm for preserving useful discriminant directions on the other hand kmsda uses the kernel trick to separate data with nonlinearly separable subclass structure extensive experimentation shows that the proposed methods outperform conventional msda and other linear discriminant analysis variants

on the set of imputations induced by the k additive core an extension to the classical notion of core is the notion of k additive core that is the set of k additive games which dominate a given game where a k additive game has its mobius transform or harsanyi dividends vanishing for subsets of more than k elements therefore the 1 additive core coincides with the classical core the advantages of the k additive core is that it is never empty once k geq 2 and that it preserves the idea of coalitional rationality however it produces k imputations that is imputations on individuals and coalitions of at most k individuals instead of a classical imputation therefore one needs to derive a classical imputation from a k order imputation by a so called sharing rule the paper investigates what set of imputations the k additive core can produce from a given sharing rule

an efficient numerical method for studying interfacial motion in two dimensional creeping flows we present new methods for computing the motion of two dimensional closed interfaces in a slow viscous flow the interfacial velocity is found through the solution to an integral equation whose analytic formulation is based on complex variable theory for the biharmonic equation the numerical methods for solving the integral equations are spectrally accurate and employ a fast multipole based iterative solution procedure which requires only o n operations where n is the number of nodes in the discretization of the interface the interface is described spectrally and we use evolution equations that preserve equal spacing in arclength of the marker points a small scale decomposition is performed to extract the dominant term in the evolution of the interface and we show that this dominant term leads to a cfl type stability constraint when in an equal arclength frame this term is linear and we show that implicit time integration schemes that are explicit in fourier space can be formulated we verify this analysis through several numerical examples

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

rotation invariant kernels and their application to shape analysis shape analysis requires invariance under translation scale and rotation translation and scale invariance can be realized by normalizing shape vectors with respect to their mean and norm this maps the shape feature vectors onto the surface of a hypersphere after normalization the shape vectors can be made rotational invariant by modeling the resulting data using complex scalar rotation invariant distributions defined on the complex hypersphere e g using the complex bingham distribution however the use of these distributions is hampered by the difficulty in estimating their parameters and the nonlinear nature of their formulation in the present paper we show how a set of kernel functions that we refer to as rotation invariant kernels can be used to convert the original nonlinear problem into a linear one as their name implies these kernels are defined to provide the much needed rotation invariance property allowing one to bypass the difficulty of working with complex spherical distributions the resulting approach provides an easy fast mechanism for 2d 3d shape analysis extensive validation using a variety of shape modeling and classification problems demonstrates the accuracy of this proposed approach

pruning noisy bases in discriminant analysis the success of linear discriminant analysis lda is due in part to the simplicity of its formulation which reduces to a simultaneous diagonalization of two symmetric matrices a and b however a fundamental drawback of this approach is that it cannot be efficiently applied wherever the matrix a is singular or when some of the smallest variances in are due to noise in this paper we present a factorization a 1b of and a correlation based criterion that can be readily employed to solve these problems we provide detailed derivations for the linear and nonlinear classification problems the usefulness of the proposed approach is demonstrated thoroughly using a large variety of databases

minimizing nearest neighbor classification error for nonparametric dimension reduction in this brief we show that minimizing nearest neighbor classification error mnne is a favorable criterion for supervised linear dimension reduction sldr we prove that mnne is better than maximizing mutual information in the sense of being a proxy of the bayes optimal criterion based on kernel density estimation we derive a nonparametric algorithm for mnne experiments on benchmark data sets show the superiority of mnne over existing nonparametric sldr methods

multi class discriminant kernel learning via convex programming regularized kernel discriminant analysis rkda performs linear discriminant analysis in the feature space via the kernel trick its performance depends on the selection of kernels in this paper we consider the problem of multiple kernel learning mkl for rkda in which the optimal kernel matrix is obtained as a linear combination of pre specified kernel matrices we show that the kernel learning problem in rkda can be formulated as convex programs first we show that this problem can be formulated as a semidefinite program sdp based on the equivalence relationship between rkda and least square problems in the binary class case we propose a convex quadratically constrained quadratic programming qcqp formulation for kernel learning in rkda a semi infinite linear programming silp formulation is derived to further improve the efficiency we extend these formulations to the multi class case based on a key result established in this paper that is the multi class rkda kernel learning problem can be decomposed into a set of binary class kernel learning problems which are constrained to share a common kernel based on this decomposition property sdp formulations are proposed for the multi class case furthermore it leads naturally to qcqp and silp formulations as the performance of rkda depends on the regularization parameter we show that this parameter can also be optimized in a joint framework with the kernel extensive experiments have been conducted and analyzed and connections to other algorithms are discussed

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

numerical analysis of tips in viscous flow when a drop or gas bubble is placed in a strong viscous flow e g a shear flow it develops very sharp tips at its ends sharp tips are also formed when a viscous fluid is withdrawn from the neighborhood of its interface with the ambient air or with another fluid selective withdrawal however it is observed frequently that there exists a critical flow strength above which the drop transitions toward a jetting state in which a jet comes out from the tip in this paper we look numerically for stationary drop shapes both globally and close to the tip which we study with very high resolution to this end we use a boundary integral method to solve the axisymmetric flow equations for arbitrary viscosity ratios in the inertialess stokes limit stationary states are solved for using newton s method this permits us to find both stable and unstable steady states and to investigate the nature of the jetting transition the critical parameters for this transition are in reasonable agreement with slender body theory excellent agreement is found with our earlier experiments in the selective withdrawal geometry s courrech du pont and j eggers phys rev lett 96 034501 2006 for which the viscosity of the phase inside the tip is negligible we describe a scale invariance of the experimental interface profiles away from the tip then we investigate the highly curved tip region not considered previously with comparable precision we find that the shape near the tip is universal i e independent of the outer flow and of the geometry of the system drop or selective withdrawal while the tip curvature becomes extremely large it always remains finite if surface tension is present

a small scale decomposition for 3d boundary integral computations with surface tension abstract an efficient non stiff boundary integral method for the initial value problem for interfacial darcy flow which is a model of porous media flow in three space dimensions is presented we consider a doubly periodic interface separating two fluids with surface tension present at the boundary surface tension introduces high order i e high derivative terms in the governing equation and this imposes a severe stability constraint on explicit time integration methods furthermore the high order terms appear in a nonlocal operator which makes it difficult to design an efficient implicit method the stiffness is removed by developing a small scale decomposition in the spirit of prior work in the two dimensional problem by hou lowengrub and shelley in order to develop this small scale decomposition we formulate the problem using a generalized isothermal parameterization of the free surface an additional difficulty is the efficient calculation of the birkhoff rott integral for the velocity of the interface we present a new algorithm based on ewald summation to compute this in o n log n operations where n is the number of interface grid points our non stiff method is expected to apply widely to problems for doubly periodic interfacial flow with surface tension which have a boundary integral formulation

multiobjective optimization for model selection in kernel methods in regression regression plays a major role in many scientific and engineering problems the goal of regression is to learn the unknown underlying function from a set of sample vectors with known outcomes in recent years kernel methods in regression have facilitated the estimation of nonlinear functions however two major interconnected problems remain open the first problem is given by the bias versus variance tradeoff if the model used to estimate the underlying function is too flexible i e high model complexity the variance will be very large if the model is fixed i e low complexity the bias will be large the second problem is to define an approach for selecting the appropriate parameters of the kernel function to address these two problems this paper derives a new smoothing kernel criterion which measures the roughness of the estimated function as a measure of model complexity then we use multiobjective optimization to derive a criterion for selecting the parameters of that kernel the goal of this criterion is to find a tradeoff between the bias and the variance of the learned function that is the goal is to increase the model fit while keeping the model complexity in check we provide extensive experimental evaluations using a variety of problems in machine learning pattern recognition and computer vision the results demonstrate that the proposed approach yields smaller estimation errors as compared with methods in the state of the art

boundary integral techniques for multi connected domains abstract several boundary integral techniques are available for the computation of the solution to laplace s equation in multi connected domains however for cases where the domain is changing such as incompressible inviscid fluid flow with free surfaces iterative methods are highly attractive the paper describes one such formulation and tests it on circular and elliptic annuli it is necessary to use interpolated quadrature points to maintain accuracy when regions of the annuli are thin

fluid dynamic models for bhattacharyya based discriminant analysis classical discriminant analysis attempts to discover a low dimensional subspace where class label information is maximally preserved under projection canonical methods for estimating the subspace optimize an information theoretic criterion that measures the separation between the class conditional distributions unfortunately direct optimization of the information theoretic criteria is generally non convex and intractable in high dimensional spaces in this work we propose a novel tractable algorithm for discriminant analysis that considers the class conditional densities as interacting fluids in the high dimensional embedding space we use the bhattacharyya criterion as a potential function that generates forces between the interacting fluids and derive a computationally tractable method for finding the low dimensional subspace that optimally constrains the resulting fluid flow we show that this model properly reduces to the optimal solution for homoscedastic data as well as for heteroscedastic gaussian distributions with equal means we also extend this model to discover optimal filters for discriminating gaussian processes and provide experimental results and comparisons on a number of datasets

stability analysis of finite difference pseudospectral and fourier galerkin approximations for time dependent problems we consider finite difference pseudospectral and fourier galerkin methods for the approximate solution of time dependent problems the paper provides a unified framework for the stability analysis of all three discrete methods in particular the problem of stability for highly accurate stencils is studied in some detail

multiple ordinal regression by maximizing the sum of margins human preferences are usually measured using ordinal variables a system whose goal is to estimate the preferences of humans and their underlying decision mechanisms requires to learn the ordering of any given sample set we consider the solution of this ordinal regression problem using a support vector machine algorithm specifically the goal is to learn a set of classifiers with common direction vectors and different biases correctly separating the ordered classes current algorithms are either required to solve a quadratic optimization problem which is computationally expensive or based on maximizing the minimum margin i e a fixed margin strategy between a set of hyperplanes which biases the solution to the closest margin another drawback of these strategies is that they are limited to order the classes using a single ranking variable e g perceived length in this paper we define a multiple ordinal regression algorithm based on maximizing the sum of the margins between every consecutive class with respect to one or more rankings e g perceived length and weight we provide derivations of an efficient easy to implement iterative solution using a sequential minimal optimization procedure we demonstrate the accuracy of our solutions in several data sets in addition we provide a key application of our algorithms in estimating human subjects ordinal classification of attribute associations to object categories we show that these ordinal associations perform better than the binary one typically employed in the literature

axisymmetric vortex sheet motion accurate evaluation of the principal value integral this paper concerns the accurate evaluation of the principal value integral governing axisymmetric vortex sheet motion previous quadrature rules for this integral lose accuracy near the axis of symmetry an approximation by de bernadinis and moore dbm that converges pointwise at the rate of o h3 has maximal errors near the axis that are o h as a result the discretization error is not smooth it contains high wavenumber frequencies that make it difficult to resolve the vortex sheet motion this paper explains the reason for the degeneracy near the axis and proposes a modified quadrature rule that is uniformly o h3 the results are based on an analytic approximation of the integrand whose integral can be precomputed the modification is implemented at negligible additional cost per timestep as an example it is applied to compute the evolution of an initially spherical vortex sheet

a boundary integral approach to unstable solidification we consider the supercooled stefan problem with a general anisotropic curvature and velocity dependent boundary condition on the moving interface we present numerical methods based on an integral equation formulation and including a new algorithm for moving curves with curvature dependent velocity these methods compute a periodic interface with o t accuracy where t is the time step previous work has been limited to short time spans and achieved slightly less than o t12 accuracy accurate numerical results are seen to agree with the predictions of linear stability theory this agreement has eluded previous authors because their numerical methods suffered from grid effects and their linear stability theory was incorrect we study the long time evolution of an unstable interface our computations exhibit the beginnings of a sidebranching instability when the boundary condition includes anisotropy and tip splitting in the isotropic case

the immersed interface method for axis symmetric problems and application to the hele shaw flow many physical application problems are axis symmetric using axis symmetric properties many three dimensional problems can be solved efficiently using two dimensional axis symmetric coordinates in this paper the immersed interface method iim in axis symmetric coordinates is developed for elliptic interface problems that have a discontinuous coefficient solution or flux a staggered grid is used to overcome the pole singularity other challenges include deriving the jump relations in axis symmetric coordinates designing the numerical algorithm when the interface is close to the pole r 0 computing interface quantities such as the normal and tangential directions surface derivatives curvature information etc the numerical algorithm is based on a finite difference discretization and uniform grid in the axis symmetric coordinates the finite difference scheme is the standard one away from the interface but is modified at grid points near and on the interface the method is shown to be second order accurate in the infinity norm the developed new iim is applied to the hele shaw flow and compared with the results from the literature

subclass graph embedding and a marginal fisher analysis paradigm subspace learning techniques have been extensively used for dimensionality reduction dr in many pattern classification problem domains recently methods like subclass discriminant analysis sda and clustering based discriminant analysis cda which use subclass information for the discrimination between the data classes have attracted much attention in parallel important work has been accomplished on graph embedding ge which is a general framework unifying several subspace learning techniques in this paper ge has been extended in order to integrate subclass discriminant information resulting to the novel subclass graph embedding sge framework which is the main contribution of our work it is proven that sge encapsulates a diversity of both supervised and unsupervised unimodal methods like locality preserving projections lpp principal component analysis pca and linear discriminant analysis lda the theoretical link of sda and cda methods with sge is also established along these lines it is shown that sge comprises a generalization of the typical ge framework including subclass dr methods moreover it allows for an easy utilization of kernels for confronting non linearly separable data employing sge in this paper a novel dr algorithm which uses subclass discriminant information called subclass marginal fisher analysis smfa has been proposed through a series of experiments on various real world datasets it is shown that smfa outperforms in most of the cases the state of the art demonstrating the efficacy and power of sge as a platform to develop new methods highlightsgraph embedding is extended in order to integrate subclass informationthe novel subclass graph embedding framework is proposed the kernelized version of the new framework is presentedsubclass graph embedding encapsulates various subspace learning methods a novel subclass marginal fisher analysis method is proposed

kernel optimization in discriminant analysis kernel mapping is one of the most used approaches to intrinsically derive nonlinear classifiers the idea is to use a kernel function which maps the original nonlinearly separable problem to a space of intrinsically larger dimensionality where the classes are linearly separable a major problem in the design of kernel methods is to find the kernel parameters that make the problem linear in the mapped representation this paper derives the first criterion that specifically aims to find a kernel representation where the bayes classifier becomes linear we illustrate how this result can be successfully applied in several kernel discriminant analysis algorithms experimental results using a large number of databases and classifiers demonstrate the utility of the proposed approach the paper also shows theoretically and experimentally that a kernel version of subclass discriminant analysis yields the highest recognition rates

application of adaptive quadrature to axi symmetric vortex sheet motion studies of the formation of fine structures on free surfaces in liquids such as curvature singularities or interface pinching demand that the motion of the interface must be computed very accurately boundary integral techniques are a popular choice in such studies because they reduce the dimension of the problem by one on the other hand the boundary integrals are singular and their accurate evaluation can prove quite challenging in two dimensional motion the interface is just a curve when this curve is closed or periodic the singularity in the integrand may be removed and the trapezoidal rule may be applied with spectral accuracy unfortunately the nature of the singularity in the integrand for three dimensional motion is much more difficult to treat in this paper we present an accurate adaptive quadrature to compute the motion of a vortex sheet in axi symmetric flow the technique is based on a vector potential formulation which offers some computational advantages over other methods based on the biot savart integral direct numerical computations show that our technique is much more accurate and efficient than existing techniques we apply our technique to study the evolution of an initially spherical vortex sheet we present evidence of the formation of a 3 2 power singularity in the curvature of the vortex sheet

max min distance analysis by using sequential sdp relaxation for dimension reduction we propose a new criterion for discriminative dimension reduction max min distance analysis mmda given a data set with c classes represented by homoscedastic gaussians mmda maximizes the minimum pairwise distance of these c classes in the selected low dimensional subspace thus unlike fisher s linear discriminant analysis flda and other popular discriminative dimension reduction criteria mmda duly considers the separation of all class pairs to deal with general case of data distribution we also extend mmda to kernel mmda kmmda dimension reduction via mmda kmmda leads to a nonsmooth max min optimization problem with orthonormal constraints we develop a sequential convex relaxation algorithm to solve it approximately to evaluate the effectiveness of the proposed criterion and the corresponding algorithm we conduct classification and data visualization experiments on both synthetic data and real data sets experimental results demonstrate the effectiveness of mmda kmmda associated with the proposed optimization algorithm

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

kpca plus lda a complete kernel fisher discriminant framework for feature extraction and recognition this paper examines the theory of kernel fisher discriminant analysis kfd in a hilbert space and develops a two phase kfd framework i e kernel principal component analysis kpca plus fisher linear discriminant analysis lda this framework provides novel insights into the nature of kfd based on this framework the authors propose a complete kernel fisher discriminant analysis ckfd algorithm ckfd can be used to carry out discriminant analysis in double discriminant subspaces the fact that it can make full use of two kinds of discriminant information regular and irregular makes ckfd a more powerful discriminator the proposed algorithm was tested and evaluated using the feret face database and the cenparmi handwritten numeral database the experimental results show that ckfd outperforms other kfd algorithms

computation of axisymmetric suction flow through porous media in the presence of surface tension abstract the effect of small surface tension on a class of axisymmetric flows with suction is studied numerically the dynamic evolution of a blob of incompressible viscous fluid surrounded by air and drawn into an interior sink is considered the velocity field of the viscous fluid is assumed to satisfy darcy s law and thus the motion is that of a flow through porous media the fluid interface motion is computed using a highly accurate boundary integral method this method combines recent numerical techniques to achieve efficient and high order space and time discretizations for this type of flow through accurate computations it is shown that in the presence of small surface tension the dynamic behavior of the axisymmetric flows is very similar to that of the two dimensional hele shaw counterparts a long finger develops in the fluid interface and forms a cone singularity when it reaches the sink before all the fluid is sucked out the finger bulges and develops a neck for sufficiently small surface tension the bulge neck formation is enhanced by the additional component of the mean curvature in the three dimensional flow however its effect is not strong enough to cause the interface to pinch off at the neck for the data considered here

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

geometry based ensembles toward a structural characterization of the classification boundary this article introduces a novel binary discriminative learning technique based on the approximation of the non linear decision boundary by a piece wise linear smooth additive model the decision border is geometrically defined by means of the characterizing boundary points points that belong to the optimal boundary under a certain notion of robustness based on these points a set of locally robust linear classifiers is defined and assembled by means of a tikhonov regularized optimization procedure in an additive model to create a final lambda smooth decision rule as a result a very simple and robust classifier with a strong geometrical meaning and non linear behavior is obtained the simplicity of the method allows its extension to cope with some of nowadays machine learning challenges such as online learning large scale learning or parallelization with linear computational complexity we validate our approach on the uci database finally we apply our technique in online and large scale scenarios and in six real life computer vision and pattern recognition problems gender recognition intravascular ultrasound tissue classification speed traffic sign detection chagas disease severity detection clef classification and action recognition using a 3d accelerometer data the results are promising and this paper opens a line of research that deserves further attention

bayes optimality in linear discriminant analysis we present an algorithm that provides the one dimensional subspace where the bayes error is minimized for the c class problem with homoscedastic gaussian distributions our main result shows that the set of possible one dimensional spaces v for which the order of the projected class means is identical defines a convex region with associated convex bayes error function g v this allows for the minimization of the error function using standard convex optimization algorithms our algorithm is then extended to the minimization of the bayes error in the more general case of heteroscedastic distributions this is done by means of an appropriate kernel mapping function this result is further extended to obtain the d dimensional solution for any given d by iteratively applying our algorithm to the null space of the d l dimensional solution we also show how this result can be used to improve upon the outcomes provided by existing algorithms and derive a low computational cost linear approximation extensive experimental validations are provided to demonstrate the use of these algorithms in classification data analysis and visualization

discriminative transformation for multi dimensional temporal sequences feature space transformation techniques have been widely studied for dimensionality reduction in vector based feature space however these techniques are inapplicable to sequence data because the features in the same sequence are not independent in this paper we propose a method called max min inter sequence distance analysis mmsda to transform features in sequences into a low dimensional subspace such that different sequence classes are holistically separated to utilize the temporal dependencies mmsda first aligns features in sequences from the same class to an adapted number of temporal states and then constructs the sequence class separability based on the statistics of these ordered states to learn the transformation mmsda formulates the objective of maximizing the minimal pairwise separability in the latent subspace as a semi definite programming problem and provides a new tractable and effective solution with theoretical proofs by constraints unfolding and pruning convex relaxation and within class scatter compression extensive experiments on different tasks have demonstrated the effectiveness of mmsda

subclass discriminant analysis over the years many discriminant analysis da algorithms have been proposed for the study of high dimensional data in a large variety of problems each of these algorithms is tuned to a specific type of data distribution that which best models the problem at hand unfortunately in most problems the form of each class pdf is a priori unknown and the selection of the da algorithm that best fits our data is done over trial and error ideally one would like to have a single formulation which can be used for most distribution types this can be achieved by approximating the underlying distribution of each class with a mixture of gaussians in this approach the major problem to be addressed is that of determining the optimal number of gaussians per class i e the number of subclasses in this paper two criteria able to find the most convenient division of each class into a set of subclasses are derived extensive experimental results are shown using five databases comparisons are given against linear discriminant analysis lda direct lda dlda heteroscedastic lda hlda nonparametric da nda and kernel based lda k lda we show that our method is always the best or comparable to the best

singularities in hele shaw flows driven by a multipole we study analytically and numerically singularity formation in an interface flow driven by a multipole for a two dimensional hele shaw cell with surface tension our analysis proves that singularity formation is inevitable in the case of a dipole for a multipole of a higher order we show that the solution does not tend to any stationary solution as time goes to infinity if its initial center of mass is not at the multipole it is therefore very likely that this solution will develop finite time singularities our extensive numerical studies suggest that a solution develops finite time singularities via the interface reaching the multipole while forming a corner at the tip of the finger that touches the multipole in addition it is observed that the interface approaches the multipole from directions which can be predicted beforehand we also estimate as a function of time the distance between the finger tip and multipole and the results are in excellent agreement with numerical computations

a fast iterative algorithm for elliptic interface problems a fast second order accurate iterative method is proposed for the elliptic equation grad cdot beta x y grad u f x y in a rectangular region omega in two space dimensions we assume that there is an irregular interface across which the coefficient beta the solution u and its derivatives and or the source term f may have jumps we are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large the interface may or may not align with an underlying cartesian grid the idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by leveque and li siam j numer anal 4 1994 pp 1019 1044 in order to take advantage of fast poisson solvers on a rectangular region an intermediate unknown function the jump in the normal derivative across the interface is introduced our discretization is equivalent to using a second order difference scheme for a corresponding poisson equation in the region and a second order discretization for a neumann like interface condition thus second order accuracy is guaranteed a gmres iteration is employed to solve the schur complement system derived from the discretization a new weighted least squares method is also proposed to approximate interface quantities from a grid function numerical experiments are provided and analyzed the number of iterations in solving the schur complement system appears to be independent of both the jump in the coefficient and the mesh size

multiclass linear dimension reduction by weighted pairwise fisher criteria we derive a class of computationally inexpensive linear dimension reduction criteria by introducing a weighted variant of the well known k class fisher criterion associated with linear discriminant analysis lda it can be seen that lda weights contributions of individual class pairs according to the euclidean distance of the respective class means we generalize upon lda by introducing a different weighting function

on the core of multiple longest traveling salesman games in this paper we introduce multiple longest traveling salesman mlts games an mlts game arises from a network in which a salesman has to visit each node player precisely once except to his home location in such an order that maximizes the total reward first it is shown that the value of a coalition of an mlts game is determined by taking the maximum of suitable combinations of one and two person coalitions secondly it is shown that mlts games with five or less players have a nonempty core however a six player mlts game may have an empty core for the special instance in which the reward between a pair of nodes is equal to 0 or 1 we provide relations between the structure of the core and the underlying network

generalized discriminant analysis using a kernel approach we present a new method that we call generalized discriminant analysis gda to deal with nonlinear discriminant analysis using kernel function operator the underlying theory is close to the support vector machines svm insofar as the gda method provides a mapping of the input vectors into high dimensional feature space in the transformed space linear properties make it easy to extend and generalize the classical linear discriminant analysis lda to nonlinear discriminant analysis the formulation is expressed as an eigenvalue problem resolution using a different kernel one can cover a wide class of nonlinearities for both simulated data and alternate kernels we give classification results as well as the shape of the decision function the results are confirmed using real data to perform seed classification

singularities in hele shaw flows we study analytically and numerically singularities in an interface flow driven by surface tension and a sink in a two dimensional hele shaw cell our analysis proves that singularity formation is inevitable in generic situations our numerical simulations suggest that the hele shaw solution develops singularities via the interface reaching the sink before all the fluid is sucked out a corner is formed right at the tip of the interface that touches the sink

integrated fisher linear discriminants this paper studies fisher linear discriminants flds based on classification accuracies for imbalanced datasets an optimal threshold is found out from a series of empirical formulas developed which is related not only to sample sizes but also to distribution regions a mixed binary decimal coding system is suggested to make the very dense datasets sparse and enlarge the class margins on condition that the neighborhood relationships of samples are nearly preserved the within class scatter matrices being or approximately singular should be moderately reduced in dimensionality but not added with tiny perturbations the weight vectors can be further updated by a kind of epoch limited three at most iterative learning strategy provided that the current training error rates come down accordingly putting the above ideas together this paper proposes a type of integrated flds the extensive experimental results over real world datasets have demonstrated that the integrated flds have obvious advantages over the conventional flds in the aspects of learning and generalization performances for the imbalanced datasets an optimal threshold is found out from a series of empirical threshold formulas developed for fisher linear discriminants based on classification accuracies weight vectors and thresholds are updated by an epoch limited iterative learning strategy the singular within class scatter matrices are reduced in dimensionality but not added with perturbations a coding system enlarges class margins and approximately preserves neighborhood relationships an integrated learning algorithm improves the learning and generalization performances of fisher linear discriminants

traveling salesman games in this paper we discuss the problem of how to divide the total cost of a round trip along several institutes among the institutes visited we introduce two types of cooperative games fixed route traveling salesman games and traveling salesman games as a tool to attack this problem under very mild conditions we prove that fixed route traveling salesman games have non empty cores if the fixed route is a solution of the classical traveling salesman problem core elements provide us with fair cost allocations a traveling salesman game may have an empty core even if the cost matrix satisfies the triangle inequality in this paper we introduce a class of matrices defining ts games with non empty cores

nonparametric discriminant analysis a nonparametric method of discriminant analysis is proposed it is based on nonparametric extensions of commonly used scatter matrices two advantages result from the use of the proposed nonparametric scatter matrices first they are generally of full rank this provides the ability to specify the number of extracted features desired this is in contrast to parametric discriminant analysis which for an l class problem typically can determine at most l 1 features second the nonparametric nature of the scatter matrices allows the procedure to work well even for non gaussian data sets using the same basic framework a procedure is proposed to test the structural similarity of two distributions the procedure works in high dimensional space it specifies a linear decomposition of the original data space in which a relative indication of dissimilarity along each new basis vector is provided the nonparametric scatter matrices are also used to derive a clustering procedure which is recognized as a k nearest neighbor version of the nonparametric valley seeking algorithm the form which results provides a unified view of the parametric nearest mean reclassification algorithm and the nonparametric valley seeking algorithm

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

linear dimensionality reduction via a heteroscedastic extension of lda the chernoff criterion we propose an eigenvector based heteroscedastic linear dimension reduction ldr technique for multiclass data the technique is based on a heteroscedastic two class technique which utilizes the so called chernoff criterion and successfully extends the well known linear discriminant analysis lda the latter which is based on the fisher criterion is incapable of dealing with heteroscedastic data in a proper way for the two class case the between class scatter is generalized so to capture differences in co variances it is shown that the classical notion of between class scatter can be associated with euclidean distances between class means from this viewpoint the between class scatter is generalized by employing the chernoff distance measure leading to our proposed heteroscedastic measure finally using the results from the two class case a multiclass extension of the chernoff criterion is proposed this criterion combines separation information present in the class mean as well as the class covariance matrices extensive experiments and a comparison with similar dimension reduction techniques are presented

two stage multiple kernel learning for supervised dimensionality reduction in supervised dimensionality reduction methods for pattern recognition tasks the information of the class labels is considered through the process of reducing the input dimensionality to improve the classification accuracy using nonlinear mappings for this purpose makes these models more appropriate for nonlinearly distributed data in this paper a new nonlinear supervised dimensionality reduction model is introduced the dimensionality reduction process in this model is performed through a multiple kernel learning paradigm in two stages in the first stage three suitable criteria for supervised dimensionality reduction containing fisher homoscedasticity and between class distance criteria are used to find the kernel weights with these weights a linear combination of several valid kernels is computed to make a new suitable kernel function in the second stage the kernel discriminant analysis method is employed for nonlinear supervised dimensionality reduction using the kernel computed in the first stage many experiments on a variety of real world datasets including handwritten digits images objects images and other datasets show that the proposed approach among a number of well known related techniques results in accurate and fast classifications a novel approach for non linear supervised dimensionality reduction is proposed the proposed method uses kernel discriminant analysis method with new kernels a multiple kernel learning mkl paradigm produces the new kernel function proper criteria for supervised dimensionality reduction are optimized in the mkl part the proposed method is successfully evaluated using many real world datasets

semi supervised nearest mean classification through a constrained log likelihood we cast a semi supervised nearest mean classifier previously introduced by the first author in a more principled log likelihood formulation that is subject to constraints this in turn leads us to make the important suggestion to not only investigate error rates of semi supervised learners but also consider the risk they originally aim to optimize we demonstrate empirically that in terms of classification error mixed results are obtained when comparing supervised to semi supervised nearest mean classification while in terms of log likelihood on the test set the semi supervised method consistently outperforms its supervised counterpart comparisons to self learning a standard approach in semi supervised learning are included to further clarify the way in which our constrained nearest mean classifier improves over regular supervised nearest mean classification

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

a kernel induced space selection approach to model selection in klda model selection in kernel linear discriminant analysis klda refers to the selection of appropriate parameters of a kernel function and the regularizer by following the principle of maximum information preservation this paper formulates the model selection problem as a problem of selecting an optimal kernel induced space in which different classes are maximally separated from each other a scatter matrix based criterion is developed to measure the goodness of a kernel induced space and the kernel parameters are tuned by maximizing this criterion this criterion is computationally efficient and is differentiable with respect to the kernel parameters compared with the leave one out loo or k fold cross validation cv the proposed approach can achieve a faster model selection especially when the number of training samples is large or when many kernel parameters need to be tuned to tune the regularization parameter in the klda our criterion is used together with the method proposed by saadi 2004 experiments on benchmark data sets verify the effectiveness of this model selection approach

numerical simulations of inviscid capillary pinchoff inviscid capillary pinchoff is studied numerically for an axisymmetric model problem in which a bubble pinches at two points on the symmetry plane breaking into two symmetric end bubbles and a satellite bubble in between results are presented for a range of density jumps across the bubble the numerical method uses a formulation in terms of arclength and tangent angle and incorporates a new procedure to redistribute the computational points dynamically in order to maintain resolution in regions of high curvature the results are compared with alternative computations by lepinnen and lister phys fluids 15 2003 568 where available new results include details about the cone crater structure near pinchoff and the dependence of satellite bubble volume and pinchoff time on the density ratio

high order quadratures for the evaluation of interfacial velocities in axi symmetric stokes flows we propose new high order accurate methods to compute the evolution of axi symmetric interfacial stokes flow the velocity at a point on the interface is given by an integral over the surface quadrature rules to evaluate these integrals are developed using asymptotic expansions of the integrands both locally about the point of evaluation and about the poles where the interface crosses the axis of symmetry the local expansions yield methods that converge to the chosen order pointwise for fixed evaluation point the pole expansions yield corrections that remove maximal errors of low order introduced by singular behaviour of the integrands as the evaluation point approaches the poles an interesting example of roundoff error amplification due to cancellation is also addressed the result is a uniformly accurate fifth order method second order pointwise fifth order and uniform fifth order methods are applied to compute three sample flows each of which presents a different computational difficulty an initially bar belled drop that pinches in finite time a drop in a strain flow that approaches a steady state and a continuously extending drop in each case the fifth order methods significantly improve the ability to resolve the flow the examples furthermore give insight into the effect of the corrections needed for uniformity we determine conditions under which the pointwise method is sufficient to obtain resolved results and others under which the corrections significantly improve the results

a cauchy integral approach to hele shaw problems with a free boundary the case of zero surface tension in this paper we study a nonlinear and nonlocal free boundary dynamics the hele shaw problem without surface tension when the fluid domain is either bounded or unbounded the key idea is to use a global quantity the cauchy integral of the free boundary to capture the motion of the boundary this cauchy integral is shown to be linear in time the free boundary at a fixed time is then recovered from its cauchy integral at that time the main tool in our analysis ischerednichenko s theorem concerning the inverse properties of the cauchy integrals as products of our approach we establish the short time existence and uniqueness of classical solutions for analytic initial boundaries we also show the non existence of classical solutions for all smooth but non analytic initial boundaries when there is a sink at either a finite point or at infinity when the fluid domain is bounded all solutions except the circular one break down before all the fluid is sucked out from the sink regularity results are also obtained when there is a source at a finite point or at infinity

a novel locally linear knn method with applications to visual recognition a locally linear k nearest neighbor llk method is presented in this paper with applications to robust visual recognition specifically the concept of an ideal representation is first presented which improves upon the traditional sparse representation in many ways the objective function based on a host of criteria for sparsity locality and reconstruction is then optimized to derive a novel representation which is an approximation to the ideal representation the novel representation is further processed by two classifiers namely an llk based classifier and a locally linear nearest mean based classifier for visual recognition the proposed classifiers are shown to connect to the bayes decision rule for minimum error additional new theoretical analysis is presented such as the nonnegative constraint the group regularization and the computational efficiency of the proposed llk method new methods such as a shifted power transformation for improving reliability a coefficients truncating method for enhancing generalization and an improved marginal fisher analysis method for feature extraction are proposed to further improve visual recognition performance extensive experiments are implemented to evaluate the proposed llk method for robust visual recognition in particular eight representative data sets are applied for assessing the performance of the llk method for various visual recognition applications such as action recognition scene recognition object recognition and face recognition

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested