kernel wiener filter and its application to pattern recognition the wiener filter wf is widely used for inverse problems from an observed signal it provides the best estimated signal with respect to the squared error averaged over the original and the observed signals among linear operators the kernel wf kwf extended directly from wf has a problem that an additive noise has to be handled by samples since the computational complexity of kernel methods depends on the number of samples a huge computational cost is necessary for the case by using the first order approximation of kernel functions we realize kwf that can handle such a noise not by samples but as a random variable we also propose the error estimation method for kernel filters by using the approximations in order to show the advantages of the proposed methods we conducted the experiments to denoise images and estimate errors we also apply kwf to classification since kwf can provide an approximated result of the maximum a posteriori classifier that provides the best recognition accuracy the noise term in the criterion can be used for the classification in the presence of noise or a new regularization to suppress changes in the input space whereas the ordinary regularization for the kernel method suppresses changes in the feature space in order to show the advantages of the proposed methods we conducted experiments of binary and multiclass classifications and classification in the presence of noise

rates of convergence for a class of global stochastic optimization algorithms inspired and motivated by the recent advances in simulated annealing algorithms this paper analyzes the convergence rates of a class of recursive algorithms for global optimization via monte carlo methods by using perturbed liapunov function methods stability results of the algorithms are established then the rates of convergence are ascertained by examining the asymptotic properties of suitably scaled estimation error sequences

the cauchy problem for space time monopole equations in sobolev spaces we consider the initial value problem of space time monopole equations in one space dimension with initial data in sobolev space hs observing null structures of the system we prove local well posedness in almost critical space unconditional uniqueness and global existence are proved for s 0 moreover we show that the h1 sobolev norm grows at a rate of at most c exp ct2

applying experimental design and regression splines to high dimensional continuous state stochastic dynamic programming in stochastic dynamic programming sdp with continuous state and decision variables the future value function is computed at discrete points in the state spac e interpolation can be used to approximate the values of the future value function between these discrete points however for large dimensional problems the number of discrete points required to obtain a good approximation of the future value function can be prohibitively large statistical methods of experimental design and function estimation may be employed to overcome this curse of dimensionality in this paper we describe a method for estimating the future value function by multivariate adaptive regression splines mars fit over a discretization scheme based on orthogonal array oa experimental designs because orthogonal arrays only grow polynomially in the state space dimension our oa mars method is accurately able to solve higher dimensional sdp problems than previously possible to our knowledge the most efficient method published prior to this work employs tensor product cubic splines to approximate the future value function johnson et al 1993 the computational advantages of oa ma rs are demonstrated in comparisons with the method using tensor product cubic splines for applications of an inventory forecasting sdp with up to nine state variables computed on a small workstation in particular the storage of an adequate tensor product cubic spline for six dimensions exceeds the memory of our workstation and the run time for an accurate oa mars sdp solution would be at least an order of magnitude faster than using tensor product cubic splines for higher than six dimensions

regularization theory and neural networks architectures we had previously shown that regularization principles lead to approximation schemes that are equivalent to networks with one layer of hidden units called regularization networks in particular s

uniqueness of weak solutions of 1 1 dimensional wave maps in this paper we prove that finite energy distributional solutions to the cauchy problem of wave maps from 1 1 dimensional minkowsky space to any complete riemannian manifold are unique

a new class of incremental gradient methods for least squares problems the least mean squares lms method for linear least squares problems differs from the steepest descent method in that it processes data blocks one by one with intermediate adjustment of the parameter vector under optimization this mode of operation often leads to faster convergence when far from the eventual limit and to slower sublinear convergence when close to the optimal solution we embed both lms and steepest descent as well as other intermediate methods within a one parameter class of algorithms and we propose a hybrid class of methods that combine the faster early convergence rate of lms with the faster ultimate linear convergence rate of steepest descent these methods are well suited for neural network training problems with large data sets furthermore these methods allow the effective use of scaling based for example on diagonal or other approximations of the hessian matrix

global strong solution to the thirring model in critical space abstract we obtain a strong solution in charge critical space l 2 r of the thirring system and federbusch equations in one space dimension by using solution representation of the models the uniqueness is obtained for the solution l 0 t l 2 r l 4 r a decay of local charge and asymptotic behavior of the field can be shown directly

rank constrained matrix best approximation problem with respect to skew hermitian matrices in this literature we study a rank constrained matrix approximation problem in the frobenius norm minr x kbxbaf2 where k is a nonnegative integer a and x are skew hermitian matrices by using the singular value decomposition and the spectrum decomposition we derive some conditions for the existence of skew hermitian solutions and establish general forms for the skew hermitian solutions to this matrix approximation problem

numerical solution of continuous state dynamic programs using linear and spline interpolation this paper demonstrates that the computational effort required to develop numerical solutions to continuous state dynamic programs can be reduced significantly when cubic piecewise polynomial functions rather than tensor product linear interpolants are used to approximate the value function tensor product cubic splines represented in either piecewise polynomial or b spline form and multivariate hermite polynomials are considered computational savings are possible because of the improved accuracy of higher order functions and because the smoothness of higher order functions allows efficient quasi newton methods to be used to compute optimal decisions the use of the more efficient piecewise polynomial form of the spline was slightly superior to the use of hermite polynomials for the test problem and easier to program in comparison to linear interpolation use of splines in piecewise polynomial form reduced the cpu time to obtain results of equivalent accuracy by a factor of 250 330 for a stochastic

parametric projection filter for image and signal restoration a class of linear algebraic restoration filters is derived for the linear degradation model with additive noise the filters are based on an optimization criterion involving two effects first the error in the restored noiseless image compared to the original image and second the energy of additive noise passed through the restoration these two effects can be balanced with a scalar parameter for both error types explicit expressions are derived in terms of the parameter it is shown that by allowing the first error norm to grow slightly the noise energy may be considerably reduced this also has a bearing on the analysis of the behavior of other linear parametric restoration filters

error estimates for approximate optimization by the extended ritz method an alternative to the classical ritz method for approximate optimization is investigated in the extended ritz method sets of admissible solutions are approximated by their intersections with sets of linear combinations of all n tuples of functions from a given basis this alternative scheme called variable basis approximation includes functions computable by trigonometric polynomials with free frequencies free node splines neural networks and other nonlinear approximating families estimates of rates of approximate optimization by the extended ritz method are derived upper bounds on rates of convergence of suboptimal solutions to the optimal one are expressed in terms of the degree n of variable basis functions the modulus of continuity of the functional to be minimized the modulus of tikhonov well posedness of the problem and certain norms tailored to the type of basis the results are applied to convex best approximation and to kernel methods in machine learning

local existence with minimal regularity for nonlinear wave equations in this paper we consider the cauchy problem of semilinear wave equations which satisfy the null condition we prove that the problem is locally well posed in h s 1 for s in 1 3 dimensions and s in 1 2 dimensions moreover it is shown that the result in 1 2 dimensions is sharp

a simple lemma on greedy approximation in hilbert space and convergence rates for projection pursuit regression and neural network training a general convergence criterion for certain iterative sequences in hilbert space is presented for an important subclass of these sequences estimates of the rate of convergence are given under very mild assumptions these results establish an 0 1 f4n nonsampling convergence rate for projection pursuit regression and neural network training where n represents the number of ridge functions neurons or coefficients in a greedy basis expansion

an equivalence between sparse approximation and support vector machines in the first part of this paper we show a similarity between the principle of structural risk minimization principle srm vapnik 1982 and the idea of sparse approximation as defined in chen donoho and saunders 1995 and olshausen and field 1996 then we focus on two specific approximate implementations of srm and sparse approximation which have been used to solve the problem of function approximation for srm we consider the support vector machine technique proposed by v vapnik and his team at at t bell labs and for sparse approximation we consider a modification of the basis pursuit de noising algorithm proposed by chen donoho and saunders 1995 we show that under certain conditions these two techniques are equivalent they give the same solution and they require the solution of the same quadratic programming problem

convergent ritz approximations of the set of stabilizing controllers abstract a necessary and sufficient condition for linear exponential sums to be dense in l p is derived under this condition a ritz approximation converges which has been proposed to solve linear control problems state space representations of orthonormal basis functions for multivariable systems are given

nonlinear black box models in system identification mathematical foundations we discuss several aspects of the mathematical foundations of the nonlinear black box identification problem we shall see that the quality of the identification procedure is always a result of a certain trade off between the expressive power of the model we try to identify the larger the number of parameters used to describe the model the more flexible is the approximation and the stochastic error which is proportional to the number of parameters a consequence of this trade off is the simple fact that a good approximation technique can be the basis of a good identification algorithm from this point of view we consider different approximation methods and pay special attention to spatially adaptive approximants we introduce wavelet and neuron approximations and show that they are spatially adaptive then we apply the acquired approximation experience to estimation problems finally we consider some implications of these theoretical developments for the practically implemented versions of the spatially adaptive algorithms

learning with boundary conditions kernel machines traditionally arise from an elegant formulation based on measuring the smoothness of the admissible solutions by the norm in the reproducing kernel hilbert space rkhs generated by the chosen kernel it was pointed out that they can be formulated in a related functional framework in which the green s function of suitable differential operators is thought of as a kernel in this letter our own picture of this intriguing connection is given by emphasizing some relevant distinctions between these different ways of measuring the smoothness of admissible solutions in particular we show that for some kernels there is no associated differential operator the crucial relevance of boundary conditions is especially emphasized which is in fact the truly distinguishing feature of the approach based on differential operators we provide a general solution to the problem of learning from data and boundary conditions and illustrate the significant role played by boundary conditions with examples it turns out that the degree of freedom that arises in the traditional formulation of kernel machines is indeed a limitation which is partly overcome when incorporating the boundary conditions this likely holds true in many real world applications in which there is prior knowledge about the expected behavior of classifiers and regressors on the boundary

the generalized lasso in the last few years the support vector machine svm method has motivated new interest in kernel regression techniques although the svm has been shown to exhibit excellent generalization properties in many experiments it suffers from several drawbacks both of a theoretical and a technical nature the absence of probabilistic outputs the restriction to mercer kernels and the steep growth of the number of support vectors with increasing size of the training set in this paper we present a different class of kernel regressors that effectively overcome the above problems we call this approach generalized lasso regression it has a clear probabilistic interpretation can handle learning sets that are corrupted by outliers produces extremely sparse solutions and is capable of dealing with large scale problems for regression functionals which can be modeled as iteratively reweighted least squares irls problems we present a highly efficient algorithm with guaranteed global convergence this defies a unique framework for sparse regression models in the very rich class of irls models including various types of robust regression models and logistic regression performance studies for many standard benchmark datasets effectively demonstrate the advantages of this model over related approaches

minimization of error functionals over variable basis functions generalized tikhonov well posedness is investigated for the problem of minimization of error functionals over admissible sets formed by variable basis functions i e linear combinations of a fixed number of elements chosen from a given basis without a prespecified ordering for variable basis functions of increasing complexity rates of decrease of infima of error functionals are estimated upper bounds are derived on such rates which do not exhibit the curse of dimensionality with respect to the number of variables of admissible functions consequences are considered for boolean functions and decision trees

numerical solutions to the witsenhausen counterexample by approximating networks approximate solutions to the witsenhausen counterexample 1968 are derived by constraining the unknown control functions to take on fixed structures containing free parameters to be optimized such structures are given by nonlinear approximating networks i e linear combinations of parametrized basis functions that benefit by density properties in normed linear spaces this reduces the original functional problem to a nonlinear programming one which is solved via stochastic approximation the method yields lower values of the costs than the ones achieved so far in the literature and most of all provides rather a complete overview of the shapes of the optimal control functions when the two parameters that characterize the witsenhausen counterexample vary one hidden layer neural networks are chosen as approximating networks

global strong solution to a nonlinear dirac type equation in one dimension abstract this paper studies a class of nonlinear massless dirac equations in one dimension which include the equations for the massless thirring model and the massless gross neveu model under the assumptions of the initial data having small charge and being bounded the global existence of the strong solution is established the decay of the local charge is also proved

optimum sampling vectors for wiener filter noise reduction sampling is a very important and basic technique for signal processing in the case that noise is added to a signal in the sampling process we may use a reconstruction and noise reduction filter such as the wiener filter the wiener filter provides a restored signal of which the mean square error is minimized however the mean square error by the wiener filter depends on the sampling vectors we may have the freedom to construct sampling vectors we provide optimum sampling vectors under the condition that the wiener filter is used for noise reduction for two cases wherein the noise is added before after sampling the sampling vectors provided in this paper may not be practical since they are very complicated however the minimum mean square error which we provide theoretically can be used for evaluating other sampling vectors we provide all proofs of the theorems and lemmas furthermore by experimental results we show their advantages

remarks on a weighted energy estimate and its application to nonlinear wave equations in one space dimension abstract a weighted energy estimate with tangential derivatives on the light cone is applied for the cauchy problem of semilinear wave equations with the null conditions in one space dimension the well posedness and lifespan of the solutions are considered based on the vector field method

neural approximations for multistage optimal control of nonlinear stochastic systems two main approximations are used to solve a nonlinear quadratic gaussian lqg optimal control problem the control law is assigned a given structure in which a finite number of parameters have to be determined to minimize the cost function the chosen structure is that of a multilayer feedforward neural network and the control law is given a limited memory the errors resulting front both assumptions are discussed simulation results show that the proposed method may constitute a simple and effective tool for solving to a sufficient degree of accuracy optimal control problems traditionally regarded as difficult ones

global charge solutions of maxwell dirac equations in mathbb r 1 1 we prove a global well posedness in charge space of the maxwell dirac system in one space dimension by using an explicit representation of its solution in terms of initial data a decay of local charge and asymptotic behaviors of the field can be shown directly

regularization techniques and suboptimal solutions to optimization problems in learning from data various regularization techniques are investigated in supervised learning from data theoretical features of the associated optimization problems are studied and sparse suboptimal solutions are searched for rates of approximate optimization are estimated for sequences of suboptimal solutions formed by linear combinations of n tuples of computational units and statistical learning bounds are derived as hypothesis sets reproducing kernel hilbert spaces and their subsets are considered

fixed point continuation for ell_1 minimization methodology and convergence we present a framework for solving the large scale ell 1 regularized convex minimization problem min x 1 mu f x our approach is based on two powerful algorithmic ideas operator splitting and continuation operator splitting results in a fixed point algorithm for any given scalar mu continuation refers to approximately following the path traced by the optimal value of x as mu increases in this paper we study the structure of optimal solution sets prove finite convergence for important quantities and establish q linear convergence rates for the fixed point algorithm applied to problems with f x convex but not necessarily strictly convex the continuation framework motivated by our convergence results is demonstrated to facilitate the construction of practical algorithms

strong solutions to stochastic wave equations with values in riemannian manifolds let m be a d dimensional compact riemannian manifold we prove existence of a unique global strong solution of the stochastic wave equation dt tu dx xu yu tu xu w where y is a c1 smooth transformation and w is a spatially homogeneous wiener process on r whose spectral measure has finite moments up to order 2

stability and generalization we define notions of stability for learning algorithms and show how to use these notions to derive generalization error bounds based on the empirical error and the leave one out error the methods we use can be applied in the regression framework as well as in the classification one when the classifier is obtained by thresholding a real valued function we study the stability properties of large classes of learning algorithms such as regularization based algorithms in particular we focus on hilbert space regularization and kullback leibler regularization we demonstrate how to apply the results to svm for regression and classification

global solution to nonlinear dirac equation for gross neveu model in 1 1 dimensions abstract this paper studies a class of nonlinear dirac equations with cubic terms in r 1 1 which include the equations for the massive thirring model and the massive gross neveu model under the assumption that the initial data has bounded l 2 norm the global existence and the uniqueness of the strong solution in c 0 l 2 r 1 are proved

hinging hyperplanes for regression classification and function approximation a hinge function y h x consists of two hyperplanes continuously joined together at a hinge in regression prediction classification pattern recognition and noiseless function approximation use of sums of hinge functions gives a powerful and efficient alternative to neural networks with computation times several orders of magnitude less than is obtained by fitting neural networks with a comparable number of parameters a simple and effective method for finding good hinges is presented

successive galerkin approximation algorithms for nonlinear optimal and robust control nonlinear optimal control and nonlinear h infinity control are two of the most significant paradigms in nonlinear systems theory unfortunately these problems require the solution of hamilton jacobi equations which are extremely difficult to solve in practice to make matters worse approximation techniques for these equations are inherently prone to the so called curse of dimensionality while there have been many attempts to approximate these equations solutions resulting in closed loop control with well defined stability and robustness have remained elusive this paper describes a recent breakthrough in approximating the hamilton jacobi bellman and hamilton jacobi isaacs equations successive approximation and galerkin approximation methods are combined to derive a novel algorithm that produces stabilizing closed loop control laws with well defined stability regions in addition we show how the structure of the algorithm can be exploited to reduce the amount of computation from exponential to po

projection filter wiener filter and karhunen loeve subspaces in digital image restoration abstract in image processing by computer the transformation from the original continuous domain image to the degraded and sampled discrete observation image is usually modelled as a linear transformation with additive noise the relation between two types of filters the wiener filter wf and the projection filter pf for the restoration of the original image from the observation is discussed the latter is based on the same principle as pseudoinverse filtering but also suppresses the additive noise the pf and the wf are shown to be closely related under a condition depending on the degradation sampling operator and the karhunen loeve expansion for the family of original images the relation between the pf and the gauss markov estimator is also clarified

comparison of worst case errors in linear and neural network approximation sets of multivariable functions are described for which worst case errors in linear approximation are larger than those in approximation by neural networks a theoretical framework for such a description is developed in the context of nonlinear approximation by fixed versus variable basis functions comparisons of approximation rates are formulated in terms of certain norms tailored to sets of basis functions the results are applied to perceptron networks

weak solutions and development of singularities of the su 2 σ model pour toutes donnees initiales d energie finie le modele sigma o k a des solutions faibles globales en chaque dimension d espace temps

on a variational norm tailored to variable basis approximation schemes a variational norm associated with sets of computational units and used in function approximation learning from data and infinite dimensional optimization is investigated for sets gk obtained by varying a vector y of parameters in a fixed structure computational unit k y e g the set of gaussians with free centers and widths upper and lower bounds on the gk variation norms of functions having certain integral representations are given in terms of the 1 norms of the weighting functions in such representations families of functions for which the two norms are equal are described

learning from examples as an inverse problem many works related learning from examples to regularization techniques for inverse problems emphasizing the strong algorithmic and conceptual analogy of certain learning algorithms with regularization algorithms in particular it is well known that regularization schemes such as tikhonov regularization can be effectively used in the context of learning and are closely related to algorithms such as support vector machines nevertheless the connection with inverse problem was considered only for the discrete finite sample problem and the probabilistic aspects of learning from examples were not taken into account in this paper we provide a natural extension of such analysis to the continuous population case and study the interplay between the discrete and continuous problems from a theoretical point of view this allows to draw a clear connection between the consistency approach in learning theory and the stability convergence property in ill posed inverse problems the main mathematical result of the paper is a new probabilistic bound for the regularized least squares algorithm by means of standard results on the approximation term the consistency of the algorithm easily follows

initial boundary value problem for nonlinear dirac equation of gross neveu type in 1 1 dimensions abstract this paper studies an initial boundary value problem for a class of nonlinear dirac equations with cubic terms which include the equations for the massive thirring model and the massive gross neveu model under the assumptions that the initial data has bounded l 2 norm and the boundary satisfies suitable conditions the global existence and the uniqueness of the strong solution are proved

local and global solutions of chern simons gauged o 3 sigma equations in one space dimension we study an initial value problem of the chern simons gauged o 3 sigma model in one space dimension the global existence of solutions to the model is proved for high regularity initial data moreover we study low regularity local well posedness observing null forms of the system and applying bilinear estimates for wave sobolev space hs as a by product a finite energy global solution is constructed

an incremental gradient projection method with momentum term and adaptive stepsize rule we consider an incremental gradient method with momentum term for minimizing the sum of continuously differentiable functions this method uses a new adaptive stepsize rule that decreases the stepsize whenever sufficient progress is not made we show that if the gradients of the functions are bounded and lipschitz continuous over a certain level set then every cluster point of the iterates generated by the method is a stationary point in addition if the gradient of the functions have a certain growth property then the method is either linearly convergent in some sense or the stepsizes are bounded away from zero the new stepsize rule is much in the spirit of heuristic learning rules used in practice for training neural networks via backpropagation as such the new stepsize rule may suggest improvements on existing learning rules finally extension of the method and the convergence results to constrained minimization is discussed as are some implementation issues and numerical experience

control of a class of nonlinear discrete time systems using multilayer neural networks a multilayer neural network nn controller is designed to deliver a desired tracking performance for the control of a class of unknown nonlinear systems in discrete time where the system nonlinearities do not satisfy a matching condition using the lyapunov approach the uniform ultimate boundedness of the tracking error and the nn weight estimates are shown by using a novel weight updates further a rigorous procedure is provided from this analysis to select the nn controller parameters the resulting structure consists of several nn function approximation inner loops and an outer proportional derivative tracking loop simulation results are then carried out to justify the theoretical conclusions the net result is the design and development of an nn controller for strict feedback class of nonlinear discrete time systems

a neural state estimator with bounded errors for nonlinear systems a neural state estimator is described acting on discrete time nonlinear systems with noisy measurement channels a sliding window quadratic estimation cost function is considered and the measurement noise is assumed to be additive no probabilistic assumptions are made on the measurement noise nor on the initial state novel theoretical convergence results are developed for the error bounds of both the optimal and the neural approximate estimators to ensure the convergence properties of the neural estimator a minimax tuning technique is used the approximate estimator can be designed offline in such a way as to enable it to process on line any possible measure pattern almost instantly

neural networks for feedback feedforward nonlinear control systems this paper deals with the problem of designing feedback feedforward control strategies to drive the state of a dynamic system in general nonlinear so as to track any desired trajectory joining the points of given compact sets while minimizing a certain cost function in general nonquadratic due to the generality of the problem conventional methods are difficult to apply thus an approximate solution is sought by constraining control strategies to take on the structure of multilayer feedforward neural networks after discussing the approximation properties of neural control strategies a particular neural architecture is presented which is based on what has been called the linear structure preserving principle the original functional problem is then reduced to a nonlinear programming one and backpropagation is applied to derive the optimal values of the synaptic weights recursive equations to compute the gradient components are presented which generalize the classical adjoint system equations of n stage optimal control theory simulation results related to nonlinear nonquadratic problems show the effectiveness of the proposed method

on the nystrom method for approximating a gram matrix for improved kernel based learning a problem for many kernel based methods is that the amount of computation required to find the solution scales as o n3 where n is the number of training examples we develop and analyze an algorithm to compute an easily interpretable low rank approximation to an n n gram matrix g such that computations of interest may be performed more rapidly the approximation is of the form gk cwk ct where c is a matrix consisting of a small number c of columns of g and wk is the best rank k approximation to w the matrix formed by the intersection between those c columns of g and the corresponding c rows of g an important aspect of the algorithm is the probability distribution used to randomly sample the columns we will use a judiciously chosen and data dependent nonuniform probability distribution let 2 and f denote the spectral norm and the frobenius norm respectively of a matrix and let gk be the best rank k approximation to g we prove that by choosing o k e4 columns g cwk ct g gk e i 1n gii2 both in expectation and with high probability for both 2 f and for all k 0 k rank w this approximation can be computed using o n additional space and time after making two passes over the data from external storage the relationships between this algorithm other related matrix decompositions and the nystrom method from integral equation theory are discussed

dependence of computational models on input dimension tractability of approximation and optimization tasks the role of input dimension d is studied in approximating in various norms target sets of d variable functions using linear combinations of adjustable computational units results from the literature which emphasize the number n of terms in the linear combination are reformulated and in some cases improved with particular attention to dependence on d for worst case error upper bounds are given in the factorized form d n where is nonincreasing typically n n 1 2 target sets of functions are described for which the function is a polynomial some important cases are highlighted where decreases to zero as d for target functions extent e g the size of domains in rd where they are defined scale e g maximum norms of target functions and smoothness e g the order of square integrable partial derivatives may depend on d and the influence of such dimension dependent parameters on model complexity is considered results are applied to approximation and solution of optimization problems by neural networks with perceptron and gaussian radial computational units

bounds on rates of variable basis and neural network approximation the tightness of bounds on rates of approximation by feedforward neural networks is investigated in a more general context of nonlinear approximation by variable basis functions tight bounds on the worst case error in approximation by linear combinations of n elements of an orthonormal variable basis are derived

on the regularity of spherically symmetric wave maps wave maps are critical points u m n of the lagrangian u m du 2 where m is an einsteinian manifold and n a riemannian one for the case m 2 1 and u a spherically symmetric map it is shown that the solution to the cauchy problem for u with smooth initial data of arbitrary size is smooth for all time provided the target manifold n satisfies the two conditions that 1 it is either compact or there exists an orthonormal frame of smooth vectorfields on n whose structure functions are bounded and 2 there are two constants c and c such that the smallest eigenvalue and the largest eigenvalue of the second fundamental form kab of any geodesic sphere p s of radius s centered at p n satisfy s c and s a c 1 s this is proved by first analyzing the energy momentum tensor and using the second condition to show that near the first possible singularity the energy of the solution cannot concentrate and hence is small one then proves that for targets satisfying the first condition initial data of small energy imply global regularity of the solution 1993 john wiley sons inc

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection

relative karhunen loeve transform the karhunen loeve transform klt provides the best approximation for a stochastic signal under the condition that its rank is fixed it has been successfully used for data compression in communication however since the klt does not consider noise its ability to suppress noise is very poor for the optimum linear data compression in the presence of noise we propose the concept of a relative karhunen loeve transform rklt it minimizes the sum of the mean squared error between the original signal and its approximation and the mean squared error caused by a noise under the condition that its rank is fixed we also provide another type of rklt it minimizes the same sum under the condition that its rank is not greater than a fixed integer since the former type of rklt does not always exist we provide a necessary and sufficient condition under which it does exist we also provide their general forms the advantage of rklts is illustrated through computer simulations