kernel based classification using quantum mechanics this paper introduces a new nonparametric estimation approach inspired from quantum mechanics kernel density estimation associates a function to each data sample in classical kernel estimation theory the probability density function is calculated by summing up all the kernels the proposed approach assumes that each data sample is associated with a quantum physics particle that has a radial activation field around it schrodinger differential equation is used in quantum mechanics to define locations of particles given their observed energy level in our approach we consider the known location of each data sample and we model their corresponding probability density function using the analogy with the quantum potential function the kernel scale is estimated from distributions of k nearest neighbours statistics in order to apply the proposed algorithm to pattern classification we use the local hessian for detecting the modes in the quantum potential hypersurface each mode is assimilated with a nonparametric class which is defined by means of a region growing algorithm we apply the proposed algorithm on artificial data and for the topography segmentation from radar images of terrain

feature extraction by non parametric mutual information maximization we present a method for learning discriminative feature transforms using as criterion the mutual information between class labels and transformed features instead of a commonly used mutual information measure based on kullback leibler divergence we use a quadratic divergence measure which allows us to make an efficient non parametric implementation and requires no prior assumptions about class densities in addition to linear transforms we also discuss nonlinear transforms that are implemented as radial basis function networks extensions to reduce the computational complexity are also presented and a comparison to greedy feature selection is made

working set selection using functional gain for ls svm the efficiency of sequential minimal optimization smo depends strongly on the working set selection this letter shows how the improvement of smo in each iteration named the functional gain fg is used to select the working set for least squares support vector machine ls svm we prove the convergence of the proposed method and give some theoretical support for its performance empirical comparisons demonstrate that our method is superior to the maximum violating pair mvp working set selection

smo based pruning methods for sparse least squares support vector machines solutions of least squares support vector machines ls svms are typically nonsparse the sparseness is imposed by subsequently omitting data that introduce the smallest training errors and retraining the remaining data iterative retraining requires more intensive computations than training a single nonsparse ls svm in this paper we propose a new pruning algorithm for sparse ls svms the sequential minimal optimization smo method is introduced into pruning process in addition instead of determining the pruning points by errors we omit the data points that will introduce minimum changes to a dual objective function this new criterion is computationally efficient the effectiveness of the proposed method in terms of computational cost and classification accuracy is demonstrated by numerical experiments

generalized information potential criterion for adaptive system training we have previously proposed the quadratic renyi s error entropy as an alternative cost function for supervised adaptive system training an entropy criterion instructs the minimization of the average information content of the error signal rather than merely trying to minimize its energy in this paper we propose a generalization of the error entropy criterion that enables the use of any order of renyi s entropy and any suitable kernel function in density estimation it is shown that the proposed entropy estimator preserves the global minimum of actual entropy the equivalence between global optimization by convolution smoothing and the convolution by the kernel in parzen windowing is also discussed simulation results are presented for time series prediction and classification where experimental demonstration of all the theoretical concepts is presented

bubble stabilization of finite element methods for the linearized incompressible navier stokes equations abstract in this paper we discuss the stabilization via bubble functions of a finite element method for stationary linearized incompressible navier stokes equations it is shown that residual free bubbles can reproduce supg

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

a pruning method of refining recursive reduced least squares support vector regression in this paper a pruning method is proposed to refine the recursive reduced least squares support vector regression rrlssvr and its improved version irrlssvr and thus two novel algorithms prurrlssvr and pruirrlssvr are yielded this pruning method ranks support vectors by defining a contribution function to the objective function and then the support vector with the least contribution is pruned unless it is the most recently selected support vector consequently prurrlssvr and pruirrlssvr outperform rrlssvr and irrlssvr respectively in terms of the number of support vectors while not impairing the generalization performance in addition a speedup scheme is presented that reduces the computational burden of computing the contribution function to show the effectiveness and feasibility of the proposed prurrlssvr and pruirrlssvr experiments are performed on ten benchmark data sets and a gas furnace instance

pruning error minimization in least squares support vector machines the support vector machine svm is a method for classification and for function approximation this method commonly makes use of an spl epsi insensitive cost function meaning that errors smaller than spl epsi remain unpunished as an alternative a least squares support vector machine lssvm uses a quadratic cost function when the lssvm method is used for function approximation a nonsparse solution is obtained the sparseness is imposed by pruning i e recursively solving the approximation problem and subsequently omitting data that has a small error in the previous pass however omitting data with a small approximation error in the previous pass does not reliably predict what the error will be after the sample has been omitted in this paper a procedure is introduced that selects from a data set the training sample that will introduce the smallest approximation error when it will be omitted it is shown that this pruning scheme outperforms the standard one

gradient based manipulation of nonparametric entropy estimates this paper derives a family of differential learning rules that optimize the shannon entropy at the output of an adaptive system via kernel density estimation in contrast to parametric formulations of entropy this nonparametric approach assumes no particular functional form of the output density we address problems associated with quantized data and finite sample size and implement efficient maximum likelihood techniques for optimizing the regularizer we also develop a normalized entropy estimate that is invariant with respect to affine transformations facilitating optimization of the shape rather than the scale of the output density kernel density estimates are smooth and differentiable this makes the derived entropy estimates amenable to manipulation by gradient descent the resulting weight updates are surprisingly simple and efficient learning rules that operate on pairs of input samples they can be tuned for data limited or memory limited situations or modified to give a fully online implementation

extensions and generalizations of a theorem of widder and of the theory of symmetric local semigroups the theory of symmetric local semigroups due to a klein and l landau j funct anal 44 1981 121 136 is generalized to semigroups indexed by subsets of rn for n 1 the result implies a similar result of a e nussbaum j funct anal 48 1982 213 223 it is further generalized to semigroups that are symmetric local in some directions and unitary in others the results are used to give a simple proof of a devinatz s duke math j 22 1955 185 192 and n i akhiezer s the classical moment problem and some related questions hafner new york 1965 generalization of a theorem of widder concerning the representation of functions as laplace integrals this result is extended to the representation as a laplace integral of a function taking values in b r the set of bounded linear operators on a hilbert space r also a theorem is proved encompassing both the result of devinatz and akhiezer and bochner s theorem on the representation of positive definite functions as fourier integrals

kernel based equiprobabilistic topographic map formation we introduce a new unsupervised competitive learning rule the kernel based maximum entropy learning rule kmer which performs equiprobabilistic topographic map formation in regular fixed topology lattices for use with nonparametric density estimation as well as nonparametric regression analysis the receptive fields of the formal neurons are overlapping radially symmetric kernels compatible with radial basis functions rbfs but unlike other learning schemes the radii of these kernels do not have to be chosen in an ad hoc manner the radii are adapted to the local input density together with the weight vectors that define the kernel centers so as to produce maps of which the neurons have an equal probability to be active equiprobabilistic maps both an online and a batch version of the learning rule are introduced which are applied to nonparametric density estimation and regression respectively the application envisaged is blind source separation bss from nonlinear noisy mixtures

reflection positivity and monotonicity we prove general reflection positivity results on riemannian manifolds for both scalar fields and dirac fields and comment on applications to quantum field theory as another application we prove the inequality cd cn between dirichlet and neumann covariance operators on a manifold with a reflection

the method of variably scaled radial kernels for solving two dimensional magnetohydrodynamic mhd equations using two discretizations the crank nicolson scheme and the method of lines mol abstract mhd equations have many applications in physics and engineering the model is coupled equations in velocity and magnetic field and has a parameter namely hartmann the value of hartmann number plays an important role in the equations when this parameter increases using different meshless methods makes the oscillations in velocity near the boundary layers in the region of the problem in the present paper a numerical meshless method based on radial basis functions rbfs is provided to solve mhd equations for approximating the spatial variable a new approach which is introduced by bozzini et al 2015 is applied the method will be used here is based on the interpolation with variably scaled kernels the methodology of the new technique is defining the scale function c on the domain r d then the interpolation problem from the data locations x j r d transforms to the new interpolation problem in the data locations x j c x j r d 1 bozzini et al 2015 the radial kernels used in the current work are multiquadrics mq inverse quadric iq and wendland s function of course the latter one is based on compactly supported functions to discretize the time variable two techniques are applied one of them is the crank nicolson scheme and another one is based on mol the numerical simulations have been carried out on the square and elliptical ducts and the obtained numerical results show the ability of the new method for solving this problem also in appendix we provide a computational algorithm for implementing the new technique in matlab software

kernel recursive least squares tracker for time varying regression in this paper we introduce a kernel recursive least squares krls algorithm that is able to track nonlinear time varying relationships in data to this purpose we first derive the standard krls equations from a bayesian perspective including a sensible approach to pruning and then take advantage of this framework to incorporate forgetting in a consistent way thus enabling the algorithm to perform tracking in nonstationary scenarios the resulting method is the first kernel adaptive filtering algorithm that includes a forgetting factor in a principled and numerically stable manner in addition to its tracking ability it has a number of appealing properties it is online requires a fixed amount of memory and computation per time step incorporates regularization in a natural manner and provides confidence intervals along with each prediction we include experimental results that support the theory as well as illustrate the efficiency of the proposed algorithm

multi parameter local semi groups of hermetian operators abstract it is proven that if a weakly continuous multi parameter family of hermetian operators in a hilbert space whose domains are not necessarily dense but whose union is dense and which has the semi group property for the parameters belonging to an open convex set in euclidean n space with 0 a limit point then the operators have self adjoint extensions which have a joint spectral representation

construction of a unique self adjoint generator for a symmetric local semigroup abstract a definition is given of a symmetric local semigroup of unbounded operators p t 0 t t for some t 0 on a hilbert space h such that p t is eventually densely defined as t 0 it is shown that there exists a unique unbounded below self adjoint operator h on h such that p t is a restriction of e th as an application it is proven that h0 v is essentially self adjoint where e th0 is an lp contractive semigroup and v is multiplication by a real measurable function such that v l2 e and e v l1 for some e 0

comments on pruning error minimization in least squares support vector machines in this letter we comment on pruning error minimization in least squares support vector machines by b j de kruif and t j a de vries the original paper proposes a way of pruning training examples for least squares support vector machines ls svm using no regularization gamma infin this causes a problem as the derivation involves inverting a matrix that is often singular we discuss a modification of this algorithm that prunes with regularization gamma finite and nonzero and is also computationally more efficient

reconstruction of kubo martin schwinger structure from euclidean green functions the explicit form of the poisson kernel for the strip t z c 0 im z is used to derive some conditions for a state on a given c algebra dynamical system a t to be a kubo martin schwinger kms state reflection positivity and generalized periodicity of the euclidean green functions corresponding to are established and used for an analytic continuation to real time reflection positivity is also used for the construction of a cyclic and faithful representation of a t on some hilbert space a sufficient condition for the identification of the constructed dynamics with the modular group of a von neumann algebra generated by the representation is formulated in terms of the euclidean green functions

online kernel based classification using adaptive projection algorithms the goal of this paper is to derive a novel online algorithm for classification in reproducing kernel hilbert spaces rkhs by exploiting projection based adaptive filtering tools the paper brings powerful convex analytic and set theoretic estimation arguments in machine learning by revisiting the standard kernel based classification as the problem of finding a point which belongs to a closed halfspace a special closed convex set in an rkhs in this way classification in an online setting where data arrive sequentially is viewed as the problem of finding a point classifier in the nonempty intersection of an infinite sequence of closed halfspaces in the rkhs convex analysis is also used to introduce sparsification arguments in the design by imposing an additional simple convex constraint on the norm of the classifier an algorithmic solution to the resulting optimization problem where new convex constraints are added every time instant is given by the recently introduced adaptive projected subgradient method apsm which generalizes a number of well known projection based adaptive filtering algorithms such as the classical normalized least mean squares nlms and the affine projection algorithm apa under mild conditions the generated sequence of estimates enjoys monotone approximation strong convergence asymptotic optimality and a characterization of the limit point further we show that the additional convex constraint on the norm of the classifier naturally leads to an online sparsification of the resulting kernel series expansion we validate the proposed design by considering the adaptive equalization problem of a nonlinear channel and by comparing it with classical as well as with recently developed stochastic gradient descent techniques

on line em algorithm for the normalized gaussian network a normalized gaussian network ngnet moody darken 1989 is a network of local linear regression units the model softly partitions the input space by normalized gaussian functions and each local unit linearly approximates the output within the partition in this article we propose a new on line em algorithm for the ngnet which is derived from the batch em algorithm xu jordan hinton 1995 by introducing a discount factor we show that the on line em algorithm is equivalent to the batch em algorithm if a specific scheduling of the discount factor is employed in addition we show that the on line em algorithm can be considered as a stochastic approximation method to find the maximum likelihood estimator a new regularization method is proposed in order to deal with a singular input distribution in order to manage dynamic environments where the input output distribution of data changes over time unit manipulation mechanisms such as unit production unit deletion and unit division are also introduced based on probabilistic interpretation experimental results show that our approach is suitable for function approximation problems in dynamic environments we also apply our on line em algorithm to robot dynamics problems and compare our algorithm with the mixtures of experts family

continuous integral kernels for unbounded schrodinger semigroups and their spectral projections abstract by suitably extending a feynman kac formula of simon canad math soc conf proc 28 2000 317 we study one parameter semigroups generated by the negative of rather general schrodinger operators which may be unbounded from below and include a magnetic vector potential in particular a common domain of essential self adjointness for such a semigroup is specified moreover each member of the semigroup is proven to be a maximal carleman operator with a continuous integral kernel given by a brownian bridge expectation the results are used to show that the spectral projections of the generating schrodinger operator also act as carleman operators with continuous integral kernels applications to schrodinger operators with rather general random scalar potentials include a rigorous justification of an integral kernel representation of their integrated density of states a relation frequently used in the physics literature on disordered solids

analytic continuation of local representations of symmetric spaces we prove the following analytic continuation theorem which applies to any virtual representation of any symmetric space g k the problem of passing from the euclidean group to the poincare group appears first to have been addressed and solved this way by klein and landau let g be a lie group k a closed subgroup and an involutive automorphism with k as fixed point subgroup if g k m is the corresponding symmetric lie algebra we form g k im and let g denote the simply connected lie group with g as lie algebra we consider virtual representations of g on a fixed complex hilbert space h adopting the definitions due to j frohlich k osterwalder and e seiler in particular g 1 g possibly unbounded operators for g in a neighborhood of e in g we prove that every such continues analytically to a strongly continuous unitary representation of g on h our theorem extends results due to klein landau frohlich et al and others earlier for special cases previous results were known only for special g k and then only for certain

w kms structure from multi time euclidean green functions an axiomatic approach to the problem of reconstruction of a dynamics from the euclidean multi time green functions is proposed the existence of a w algebra on which the reconstructed dynamics acts as the canonical group of modular automorphisms is shown a condition under which the reconstruction from the one time and multi time green functions are equivalent is formulated

an accurate stable and efficient domain type meshless method for the solution of mhd flow problems the aim of the present paper is the development of an efficient numerical algorithm for the solution of magnetohydrodynamics flow problems for regular and irregular geometries subject to dirichlet neumann and robin boundary conditions toward this the meshless point collocation method mpcm is used for mhd flow problems in channels with fully insulating or partially insulating and partially conducting walls having rectangular circular elliptical or even arbitrary cross sections mpc is a truly meshless and computationally efficient method the maximum principle for the discrete harmonic operator in the meshfree point collocation method has been proven very recently and the convergence proof for the numerical solution of the poisson problem with dirichlet boundary conditions have been attained also additionally in the present work convergence is attained for neumann and robin boundary conditions accordingly the shape functions are constructed using the moving least squares mls approximation the refinement procedure with meshless methods is obtained with an easily handled and fully automated manner we present results for hartmann number up to 10 5 the numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the exact solution and the conventional fem and bem for the dirichlet neumann and robin boundary conditions of interior problems with simple or complex boundaries

bayesian framework for least squares support vector machine classifiers gaussian processes and kernel fisher discriminant analysis the bayesian evidence framework has been successfully applied to the design of multilayer perceptrons mlps in the work of mackay nevertheless the training of mlps suffers from drawbacks like the nonconvex optimization problem and the choice of the number of hidden units in support vector machines svms for classification as introduced by vapnik a nonlinear decision boundary is obtained by mapping the input vector first in a nonlinear way to a high dimensional kernel induced feature space in which a linear large margin classifier is constructed practical expressions are formulated in the dual space in terms of the related kernel function and the solution follows from a convex quadratic programming qp problem in least squares svms ls svms the svm problem formulation is modified by introducing a least squares cost function and equality instead of inequality constraints and the solution follows from a linear system in the dual space implicitly the least squares formulation corresponds to a regression formulation and is also related to kernel fisher discriminant analysis the least squares regression formulation has advantages for deriving analytic expressions in a bayesian evidence framework in contrast to the classification formulations used for example in gaussian processes gps the ls svm formulation has clear primal dual interpretations and without the bias term one explicitly constructs a model that yields the same expressions as have been obtained with gps for regression in this article the bayesian evidence framework is combined with the ls svm classifier formulation starting from the feature space formulation analytic expressions are obtained in the dual space on the different levels of bayesian inference while posterior class probabilities are obtained by marginalizing over the model parameters empirical results obtained on 10 public domain data sets show that the ls svm classifier designed within the bayesian evidence framework consistently yields good generalization performances

further considerations on residual free bubbles for advective diffusive equations abstract we further consider the galerkin method for advective diffusive equations in two dimensions the finite dimensional space employed is of piecewise polynomials enriched with residual free bubbles rfb we show that in general this method does not coincide with the supg method unless the piecewise polynomials are spanned by linear functions furthermore a simple stability analysis argument displays the effect of the rfb on the reduced space of piecewise polynomials which in some situations is not equivalent to streamline diffusion for bilinears

spectral representations for unbounded operators with real spectrum soit t un operateur agissant dans un espace de banach reflexif x avec domaine d t et spectre reel alors il existe une mesure spectrale e sur z la variete de semisimplicite pour t telle que 1 pour chaque b r e commute avec chaque u b x qui commute avec t 2 d t z x z r se ds x existe et appartient a z et 3 tx r se ds x pour tout x d t z

the nearly optimal petrov galerkin method for convection diffusion problems abstract the nearly optimal petrov galerkin nopg method is employed to improve finite element computation of convection dominated transport phenomena the design of the nopg method for convection diffusion is based on consideration of the advective limit nonetheless the resulting method is applicable to the entire admissible range of problem parameters an investigation of the stability properties of this method leads to a coercivity inequality the convergence features of the nopg method for convection diffusion are studied in an error analysis that is based on the stability estimates the proposed method compares favorably to the performance of an established technique on several numerical tests

the generation of weakly integrable semigroups abstract a characterization of the generators of a class of weakly integrable semigroups on a locally convex space is proved it bears a resemblance to the hille yosida phillips theorem on the generators of strongly continuous semigroups on a banach space

from the euclidean group to the poincare group via osterwalder schrader positivity given a continuous representation of the euclidean group inn 1 dimensions together with a covariant system of subspaces which satisfies osterwalder schrader positivity we construct a continuous unitary representation of the orthochronous poincare group inn 1 dimensions satisfying the spectral condition a similar result holds for the covering groups of the euclidean and poincare group

finite element method solution of electrically driven magnetohydrodynamic flow the magnetohydrodynamic mhd flow in a rectangular duct is investigated for the case when the flow is driven by the current produced by electrodes placed one in each of the walls of the duct where the applied magnetic field is perpendicular the flow is steady laminar and the fluid is incompressible viscous and electrically conducting a stabilized finite element with the residual free bubble rfb functions is used for solving the governing equations the finite element method employing the rfb functions is capable of resolving high gradients near the layer regions without refining the mesh thus it is possible to obtain solutions consistent with the physical configuration of the problem even for high values of the hartmann number before employing the bubble functions in the global problem we have to find them inside each element by means of a local problem this is achieved by approximating the bubble functions by a nonstandard finite element method based on the local problem equivelocity and current lines are drawn to show the well known behaviours of the mhd flow those are the boundary layer formation close to the insulated walls for increasing values of the hartmann number and the layers emanating from the endpoints of the electrodes the changes in direction and intensity with respect to the values of wall inductance are also depicted in terms of level curves for both the velocity and the induced magnetic field

local radial point interpolation method for the fully developed magnetohydrodynamic flow abstract in this paper a local radial point interpolation method lrpim is presented to obtain the numerical solutions of the coupled equations in velocity and magnetic field for the fully developed magnetohydrodynamic mhd flow through a straight duct of rectangular section with arbitrary wall conductivity and orientation of applied magnetic field local weak forms are developed using weighted residual method locally for the governing equations of fully developed mhd flow the shape functions from lrpim possess the delta function property therefore essential boundary conditions can be applied as easily as that in the finite element method the implementation procedure of lrpim method is node based and it doesn t need any mesh or element computations have been carried out for different hartmann numbers wall conductivities and orientations of applied magnetic field

maximum margin clustering made practical motivated by the success of large margin methods in supervised learning maximum margin clustering mmc is a recent approach that aims at extending large margin methods to unsupervised learning however its optimization problem is nonconvex and existing mmc methods all rely on reformulating and relaxing the nonconvex optimization problem as semidefinite programs sdp though sdp is convex and standard solvers are available they are computationally very expensive and only small data sets can be handled to make mmc more practical we avoid sdp relaxations and propose in this paper an efficient approach that performs alternating optimization directly on the original nonconvex problem a key step to avoid premature convergence in the resultant iterative procedure is to change the loss function from the hinge loss to the laplacian square loss so that overconfident predictions are penalized experiments on a number of synthetic and real world data sets demonstrate that the proposed approach is more accurate much faster hundreds to tens of thousands of times faster and can handle data sets that are hundreds of times larger than the largest data set reported in the mmc literature

recursive reduced least squares support vector regression combining reduced technique with iterative strategy we propose a recursive reduced least squares support vector regression the proposed algorithm chooses the data which make more contribution to target function as support vectors and it considers all the constraints generated by the whole training set thus it acquires less support vectors the number of which can be arbitrarily predefined to construct the model with the similar generalization performance in comparison with other methods our algorithm also gains excellent parsimoniousness numerical experiments on benchmark data sets confirm the validity and feasibility of the presented algorithm in addition this algorithm can be extended to classification

on virtual representations of symmetric spaces and their analytic continuation in this paper we study a problem in group representation theory motivated by relativistic quantum field theory one of the most powerful approaches to constructing models of relativistic quantum fields relies on euclidean field theory a description of quantum field theory in which time is purely imaginary the basic objects of euclidean field theory the euclidean green s or schwinger functions can often be expressed in terms of functional integrals the construction of quantum field models is thereby reduced to quadratures a well known simple example of this method is the reformulation of quantum mechanisms by means of wiener integrals the fundamental solution of the schrodinger equation analytically continued to imaginary time is given by the well known feynman kac formula see e g 15 19 in this approach one faces the problem of analytic continuation back to real time while this problem is elementary in non relativistic quantum mechanics it is rather intricate in relativistic quantum field theory for some class of relativistic quantum field theories a fairly general solution to this problem was given in papers by osterwalder and

stochastically positive structures on weyl algebras the case of quasi free states we consider quasi free stochastically positive ground and thermal states on weyl algebras in the imaginary time formulation in particular we obtain a new derivation of a general form of thermal quasi free state and give conditions when such a state is stochastically positive i e when it defines a periodic stochastic process with respect to imaginary time a so called thermal process then we show that the thermal process completely determines modular structure canonically associated with the quasi free thermal state on weyl algebra we discuss a variety of examples connected with free quantum field theories on globally hyperbolic stationary space times and models of quantum statistical mechanics

evolution strategies based adaptive lp ls svm not only different databases but two classes of data within a database can also have different data structures svm and ls svm typically minimize the empirical f risk regularized versions subject to fixed penalty l 2 or l 1 penalty are non adaptive since their penalty forms are pre determined they often perform well only for certain types of situations for example ls svm with l 2 penalty is not preferred if the underlying model is sparse this paper proposes an adaptive penalty learning procedure called evolution strategies es based adaptive l p least squares support vector machine es based l p ls svm to address the above issue by introducing multiple kernels a l p penalty based nonlinear objective function is derived the iterative re weighted minimal solver irms algorithm is used to solve the nonlinear function then evolution strategies es is used to solve the multi parameters optimization problem penalty parameterp kernel and regularized parameters are adaptively selected by the proposed es based algorithm in the process of training the data which makes it easier to achieve the optimal solution numerical experiments are conducted on two artificial data sets and six real world data sets the experiment results show that the proposed procedure offer better generalization performance than the standard svm the ls svm and other improved algorithms

spectral feature projections that maximize shannon mutual information with class labels determining optimal subspace projections that can maintain task relevant information in the data is an important problem in machine learning and pattern recognition in this paper we propose a nonparametric nonlinear subspace projection technique that maintains class separability maximally under the shannon mutual information mi criterion employing kernel density estimates for nonparametric estimation of mi makes possible an interesting marriage of kernel density estimation based information theoretic methods and kernel machines which have the ability to determine nonparametric nonlinear solutions for difficult problems in machine learning significant computational savings are achieved by translating the definition of the desired projection into the kernel induced feature space which leads to obtain analytical solution

thermal quantum fields with spatially cutoff interactions in 1 1 space time dimensions we construct interacting quantum fields in 1 1 space time dimensions representing charged or neutral scalar bosons at positive temperature and zero chemical potential our work is based on prior work by klein and landau and hoegh krohn generalized path space methods are used to add a spatially cut off interaction to the free system which is described in the araki woods representation it is shown that the interacting kms state is normal w r t the araki woods representation the observable algebra and the modular conjugation of the interacting system are shown to be identical to the ones of the free system and the interacting liouvillean is described in terms of the free liouvillean and the interaction

correction to gradient based manipulation of nonparametric entropy estimates jul 04 828 837 in the above titled paper ibid vol 15 no 4 pp 828 837 jul 04 the m step 22 was incorrect the corrected m step is presented here

stochastic processes associated with kms states abstract a general mathematical framework is presented for the connection between quantum statistical mechanics and periodic stochastic processes stochastically positive kms systems periodic os positive stochastic processes periodic positive semigroup structures are defined and shown to be equivalent the objects of the tomita takesaki theory are explicitly exhibited in terms of the associated stochastic process and the statements of the theory are proven a two sided markov property for the stochastic process is related to a cyclicity property of the kms state perturbations are constructed by the feynman kac nelson formula the general framework is applied to kms states given by density matrices

a priori error analysis of residual free bubbles for advection diffusion problems we develop an a priori error analysis of a finite element approximation to the elliptic advection diffusion equation eps delta u conv cdot nabla u f subject to a homogeneous dirichlet boundary condition based on the use of residual free bubble functions an optimal order error bound is derived in the so called stability norm biggl eps nabla v 2 l 2 omega sum t h t conv cdot nabla v 2 l 2 t biggr 1 2 where ht denotes the diameter of element t in the subdivision of the computational domain

the finite element method for mhd flow at high hartmann numbers abstract a stabilized finite element method using the residual free bubble functions rfb is proposed for solving the governing equations of steady magnetohydrodynamic duct flow a distinguished feature of the rfb method is the resolving capability of high gradients near the layer regions without refining mesh we show that the rfb method is stable by proving that the numerical method is coercive even not only at low values but also at moderate and high values of the hartmann number numerical results confirming theoretical findings are presented for several configurations of interest the approximate solution obtained by the rfb method is also compared with the analytical solution of shercliff s problem

the formation of topographic maps that maximize the average mutual information of the output responses to noiseless input signals this article introduces an extremely simple and local learning rule for to pographic map formation the rule called the maximum entropy learning rule mer maximizes the unconditional entropy of the map s output for any type of input distribution the aim of this article is to show that mer is a viable strategy for building topographic maps that maximize the average mutual information of the output responses to noiseless input signals when only input noise and noise added input signals are available

fast curvature matrix vector products for second order gradient descent we propose a generic method for iteratively approximating various second order gradient steps newton gauss newton levenberg marquardt and natural gradient in linear time per iteration using special curvature matrix vector products that can be computed in o n two recent acceleration techniques for on line learning matrix momentum and stochastic meta descent smd implement this approach since both were originally derived by very different routes this offers fresh insight into their operation resulting in further improvements to smd

unitary representations of lie groups with reflection symmetry we consider the following class of unitary representations of some real lie groupgwhich has a matched pair of symmetries described as follows i supposeghas a period 2 automorphism and that the hilbert spaceh carries a unitary operatorjsuch thatj j i e selfsimilarity ii an added symmetry is implied ifh further contains a closed subspacek0having a certainorder covarianceproperty and satisfying thek0 restricted positivity v jv 0 v k0 where is the inner product inh from i ii we get an induced dual representation of an associated dual groupgc all three properties selfsimilarity order covariance and positivity are satisfied in a natural context whengis semisimple and hermitean but whengis the ax b group or the heisenberg group positivity is incompatible with the other two axioms for the infinite dimensional irreducible representations we describe a class ofg containing the latter two which admits a classification of the possible spacesk0 h satisfying the axioms of selfsimilarity and order covariance

generalized core vector machines kernel methods such as the support vector machine svm are often formulated as quadratic programming qp problems however given m training patterns a naive implementation of the qp solver takes o m 3 training time and at least o m2 space hence scaling up these qps is a major stumbling block in applying kernel methods on very large data sets and a replacement of the naive method for finding the qp solutions is highly desirable recently by using approximation algorithms for the minimum enclosing ball meb problem we proposed the core vector machine cvm algorithm that is much faster and can handle much larger data sets than existing svm implementations however the cvm can only be used with certain kernel functions and kernel methods for example the very popular support vector regression svr cannot be used with the cvm in this paper we introduce the center constrained meb problem and subsequently extend the cvm algorithm the generalized cvm algorithm can now be used with any linear nonlinear kernel and can also be applied to kernel methods such as svr and the ranking svm moreover like the original cvm its asymptotic time complexity is again linear in m and its space complexity is independent of m experiments show that the generalized cvm has comparable performance with state of the art svm and svr implementations but is faster and produces fewer support vectors on very large data sets

on the stability of residual free bubbles for convection diffusion problemsand their approximation by a two level finite element method we consider the galerkin finite element method for partial diffferential equations in two dimensions where the finite dimensional space used consists of piecewise isoparametric polynomials enriched with bubble functions writing l for the differential operator we show that for elliptic convection diffusion problems the component of the bubble enrichment that stabilizes the method is essentially equivalent to a petrov galerkin method with an l spline exponentially fitted trial space and piecewise polynomial test space the remaining component of the bubble influences the accuracy of the method a stability inequality recently obtained by brezzi franca and russo for a limiting case of bubbles applied to convection diffusion problems is shown to be slightly weaker than the standard stability inequality that is obtained for the sdfem supg method thereby demonstrating that the bubble approach is in general slightly less stable than the streamline diffusion method when the trial functions are piecewise linear we show that residual free bubbles are as stable as sdfem supg and we extend this stability inequality to include positive mesh peclet numbers in the convection dominated regime approximate computations of the residual free bubbles are performed using a two level finite element method

reflection positivity and conformal symmetry abstract a reflection positive hilbert space is a triple e e where e is a hilbert space a unitary involution and e a closed subspace on which the hermitian form v w v w is positive semidefinite for a triple g s where g is a lie group an involutive automorphism of g and s a subsemigroup invariant under the involution s s s 1 a unitary representation of g on e e is called reflection positive if g g and s e e this is the first in a series of papers in which we develop a new and systematic approach to reflection positive representations based on reflection positive distributions and reflection positive distribution vectors this approach is most natural to obtain classification results in particular in the abelian case among the tools we develop is a generalization of the bochner schwartz theorem to positive definite distributions on open convex cones we further illustrate our techniques with a non abelian example by constructing reflection positive distribution vectors for complementary series representations of the conformal group o 1 n 1 r of the sphere s n

an improved conjugate gradient scheme to the solution of least squares svm the least square support vector machines ls svm formulation corresponds to the solution of a linear system of equations several approaches to its numerical solutions have been proposed in the literature in this letter we propose an improved method to the numerical solution of ls svm and show that the problem can be solved using one reduced system of linear equations compared with the existing algorithm for ls svm the approach used in this letter is about twice as efficient numerical results using the proposed method are provided for comparisons with other existing algorithms

a bubble stabilized least squares finite element method for steady mhd duct flow problems at high hartmann numbers in this paper we devise a stabilized least squares finite element method using the residual free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow we convert the original system of second order partial differential equations into a first order system formulation by introducing two additional variables then the least squares finite element method using c 0 linear elements enriched with the residual free bubble functions for all unknowns is applied to obtain approximations to the first order system the most advantageous features of this approach are that the resulting linear system is symmetric and positive definite and it is capable of resolving high gradients near the layer regions without refining the mesh thus this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the hartmann number before incoorperating the bubble functions into the global problem we apply the galerkin least squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements therefore we indeed introduce a two level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual free bubble functions numerical results confirming theoretical findings are presented for several examples including the shercliff problem

smo algorithm for least squares svm formulations this article extends the well known smo algorithm of support vector machines svms to least squares svm formulations that include ls svm classification kernel ridge regression and a particular form of regularized kernel fisher discriminant the algorithm is shown to be asymptotically convergent it is also extremely easy to implement computational experiments show that the algorithm is fast and scales efficiently quadratically as a function of the number of examples

diffusion semigroups in spaces of continuous functions with mixed topology abstract we study transition semigroups and kolmogorov equations corresponding to stochastic semilinear equations on a hilbert space h it is shown that the transition semigroup is strongly continuous and locally equicontinuous in the space of polynomially increasing continuous functions on h when endowed with the so called mixed topology as a result we characterize cores of certain second order differential operators in such spaces and show that they have unique extensions to generators of strongly continuous semigroups

fast sparse approximation for least squares support vector machine in this paper we present two fast sparse approximation schemes for least squares support vector machine ls svm named fsals svm and pfsals svm to overcome the limitation of ls svm that it is not applicable to large data sets and to improve test speed fsals svm iteratively builds the decision function by adding one basis function from a kernel based dictionary at one time the process is terminated by using a flexible and stable epsilon insensitive stopping criterion a probabilistic speedup scheme is employed to further improve the speed of fsals svm and the resulting classifier is named pfsals svm our algorithms are of two compelling features low complexity and sparse solution experiments on benchmark data sets show that our algorithms obtain sparse classifiers at a rather low cost without sacrificing the generalization performance

weakly integrable semigroups on locally convex spaces abstract a class of weakly integrable semigroups on locally convex spaces is introduced and studied the results are illustrated by examples of semigroups of unbounded operators on a banach space which include fractional powers of a closed operator and spectral local semigroups

finite element method for solving mhd flow in a rectangular duct a finite element method is given to obtain the solution in terms of velocity and induced magnetic field for the steady mhd magnetohydrodynamic flow through a rectangular pipe having arbitrarily conducting walls linear and then quadratic approximations have been taken for both velocity and magnetic field for comparison and it is found that with the quadratic approximation it is possible to increase the conductivity and hartmann number m m 100 a special solution procedure has been used for the resulting block tridiagonal system of equations computations have been carried out for several values of hartmann number 5 m 100 and wall conductivity it is also found that if the wall conductivity increases the flux decreases the same is the effect of increasing the hartmann number selected graphs are given showing the behaviour of the velocity field and induced magnetic field

fundamental solution for coupled magnetohydrodynamic flow equations in this paper a fundamental solution for the coupled convection diffusion type equations is derived the boundary element method bem application then is established with this fundamental solution for solving the coupled equations of steady magnetohydrodynamic mhd duct flow in the presence of an external oblique magnetic field thus it is possible to solve mhd duct flow problems with the most general form of wall conductivities and for large values of hartmann number the results for velocity and induced magnetic field is visualized in terms of graphics for values of hartmann number m 300

an exponential compact difference scheme for solving 2d steady magnetohydrodynamic mhd duct flow problems in this article an exponential high order compact ehoc difference scheme on the nine point stencil is developed for the solution of the coupled equations representing the steady incompressible viscous magnetohydrodynamic mhd flow through a straight channel of rectangular section a key property of the ehoc scheme is that it has excellent stability and higher accuracy so that the high gradients near the boundary layer areas can be effectively resolved without refining the mesh numerical experiments are carried out to validate the performance of the currently proposed scheme computation results of the mhd flow in the 2d square channel problems with different wall conductivities are presented for hartmann numbers ranging from 10 to 10 6 the numerical solutions obtained with the newly developed ehoc scheme are also compared with analytic solutions and numerical results by other available methods in the literature

semilinear predictability minimization produces well known feature detectors predictability minimization pm schmidhuber 1992 exhibits various intuitive and theoretical advantages over many other methods for unsupervised redundancy reduction so far however there have not been any serious practical applications of pm in this paper we apply semilinear pm to static real world images and find that without a teacher and without any significant preprocessing the system automatically learns to generate distributed representations based on well known feature detectors such as orientation sensitive edge detectors and off center on surround detectors thus extracting simple features related to those considered useful for image preprocessing and compression

density dependent quantized least squares support vector machine for large data sets based on the knowledge that input data distribution is important for learning a data density dependent quantization scheme dqs is proposed for sparse input data representation the usefulness of the representation scheme is demonstrated by using it as a data preprocessing unit attached to the well known least squares support vector machine ls svm for application on big data sets essentially the proposed dqs adopts a single shrinkage threshold to obtain a simple quantization scheme which adapts its outputs to input data density with this quantization scheme a large data set is quantized to a small subset where considerable sample size reduction is generally obtained in particular the sample size reduction can save significant computational cost when using the quantized subset for feature approximation via the nystrom method based on the quantized subset the approximated features are incorporated into ls svm to develop a data density dependent quantized ls svm dqls svm where an analytic solution is obtained in the primal solution space the developed dqls svm is evaluated on synthetic and benchmark data with particular emphasis on large data sets extensive experimental results show that the learning machine incorporating dqs attains not only high computational efficiency but also good generalization performance

kernel least squares models using updates of the pseudoinverse sparse nonlinear classification and regression models in reproducing kernel hilbert spaces rkhss are considered the use of mercer kernels and the square loss function gives rise to an overdetermined linear least squares problem in the corresponding rkhs when we apply a greedy forward selection scheme the least squares problem may be solved by an order recursive update of the pseudoinverse in each iteration step the computational time is linear with respect to the number of the selected training samples

community detection in complex networks by density based clustering abstract we proposed a method to find the community structure in a complex network by density based clustering physical topological distance is introduced in density based clustering for determining a distance function of specific influence functions according to the distribution of the data the community structures are uncovered the method keeps a better connection mode of the community structure than the existing algorithms in terms of modularity which can be viewed as a basic characteristic of community detection in the future moreover experimental results indicate that the proposed method is efficient and effective to be used for community detection of medium and large networks

two new upwind difference schemes for a coupled system of convection diffusion equations arising from the steady mhd duct flow problems in this paper we develop two new upwind difference schemes for solving a coupled system of convection diffusion equations arising from the steady incompressible mhd duct flow problem with a transverse magnetic field at high hartmann numbers such an mhd duct flow is convection dominated and its solution may exhibit localized phenomena such as boundary layers namely narrow boundary regions where the solution changes rapidly most conventional numerical schemes cannot efficiently solve the layer problems because they are lacking in either stability or accuracy in contrast the newly proposed upwind difference schemes can achieve a reasonable accuracy with a high stability and they are capable of resolving high gradients near the layer regions without refining the grid the accuracy of the first new upwind scheme is o h k and the second one improves the accuracy to o e 2 h k email protected h 2 k 2 h 3 k 3 where 0 email protected 1 email protected 1 and m is the high hartmann number numerical examples are provided to illustrate the performance of the newly proposed upwind difference schemes

an information maximization approach to blind separation and blind deconvolution we derive a new self organizing learning algorithm that maximizes the information transferred in a network of nonlinear units the algorithm does not assume any knowledge of the input distributions and is defined here for the zero noise limit under these conditions information maximization has extra properties not found in the linear case linsker 1989 the nonlinearities in the transfer function are able to pick up higher order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation this enables the network to separate statistically independent components in the inputs a higher order generalization of principal components analysis we apply the network to the source separation or cocktail party problem successfully separating unknown mixtures of up to 10 speakers we also show that a variant on the network architecture is able to perform blind deconvolution cancellation of unknown echoes and reverberation in a speech signal finally we derive dependencies of information transfer on time delays we suggest that information maximization provides a unifying framework for problems in blind signal processing

stationary parabolic anderson model and intermittency this paper is devoted to the analysis of the large time behavior of the solutions of the anderson parabolic problem frac partial u partial t kappa delta u xi x u when the potential x is a homogeneous ergodic random field on d our goal is to prove the asymptotic spatial intermittency of the solution and for this reason we analyze the large time properties of all the moments of the positive solutions this provides an extension to the continuous space d of the work done originally by gartner and molchanov in the case of the lattice d in the process of our moment analysis we show that it is possible to exhibit new asymptotic regimes by considering a special class of generalized gaussian fields interpolating continuously between the exponent 2 which is found in the case of bona fide continuous gaussian fields x and the exponent 3 2 appearing in the case of a one dimensional white noise finally we also determine the precise almost sure large time asymptotics of the positive solutions

ergodic properties of operator semigroups in general weak topologies abstract the aim of this paper is to study ergodic properties i e properties about the limit of cesaro averages of a semigroup of bounded linear operators on a banach space x which is assumed to be continuous and locally integrable in the sense of a certain general weak topology of x then the results are applied to particular examples such as locally strongly integrable semigroups their dual semigroups and the tensor product semigroup of two c 0 semigroups

sparse lssvm in primal using cholesky factorization for large scale problems for support vector machine svm learning least squares svm lssvm derived by duality lssvm d lssvm is a widely used model because it has an explicit solution one obvious limitation of the model is that the solution lacks sparseness which limits it from training large scale problems efficiently in this paper we derive an equivalent lssvm model in primal space lssvm p lssvm by the representer theorem and prove that p lssvm can be solved exactly at some sparse solutions for problems with low rank kernel matrices two algorithms are proposed for finding the sparse approximate solution of p lssvm by cholesky factorization one is based on the decomposition of the kernel matrix k as p p top with the best low rank matrix p approximately by pivoting cholesky factorization the other is based on solving p lssvm by approximating the cholesky factorization of the hessian matrix with rank one update scheme for linear learning problems theoretical analysis and experimental results support that p lssvm can give the sparsest solutions in all svm learners experimental results on some large scale nonlinear training problems show that our algorithms based on p lssvm can converge to acceptable test accuracies at very sparse solutions with a sparsity level 1 and even as little as 0 01 hence our algorithms are a better choice for large scale training problems