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

neural network based reliability analysis a comparative study abstract a study on the applicability of different kinds of neural networks for the probabilistic analysis of structures when the sources of randomness can be modeled as random variables is summarized the networks are employed as numerical devices for substituting the finite element code needed by monte carlo simulation the comparison comprehends two network types multi layer perceptrons and radial basis functions classifiers cost functions sum of square errors and cross entropy optimization algorithms back propagation gauss newton newton raphson sampling methods for generating the training population using uniform and actual distributions of the variables and purposes of neural network use as functional approximators and data classifiers the comparative study is performed over four examples corresponding to different types of the limit state function and structural behaviors the analysis indicates some recommended ways of employing neural networks in this field

the linear algebra of the generalized pascal matrix abstract this paper discusses three kinds of generalized pascal matrix and generalizes the results of r brawer and m pirovino

adaptive neural control for a class of switched nonlinear systems in this paper adaptive neural control is presented for a class of switched nonlinear systems with switching jumps and uncertainties in both system models and switching signals condition on limiting variation of the lyapunov function is given for input to state asymptotic stability of switched systems with switching jumps the control objective is achieved uniformly with respect to a class of switching signals the coupled difficulties from the discrepancy between control gains and switching jumps are overcome by discontinuous adaptive neural control combined with the classical adaptive control smooth approximations of the discontinuous controls are then presented for a systematic design procedure

learning and generalization in radial basis function networks the two layer radial basis function network with fixed centers of the basis functions is analyzed within a stochastic training paradigm various definitions of generalization error are considered and two such definitions are employed in deriving generic learning curves and generalization properties both with and without a weight decay term the generalization error is shown analytically to be related to the evidence and via the evidence to the prediction error and free energy the generalization behavior is explored the generic learning curve is found to be inversely proportional to the number of training pairs presented optimization of training is considered by minimizing the generalization error with respect to the free parameters of the training algorithms finally the effect of the joint activations between hidden layer units is examined and shown to speed training

wiener s tauberian theorems for the fourier bessel transformation and uniform approximation by rbf networks of delsarte translates abstract we prove analogues of the wiener and wiener pitt tauberian theorems for the delsarte translation and the fourier bessel transformation through similar functional analytic techniques we give conditions on the kernel function or activation function for the family of radial basis function rbf neural networks obtained upon replacing the usual translation by the delsarte one with not necessarily the same smoothing factor in all kernel nodes to have the universal approximation property in a certain space of continuous functions endowed with the uniform norm

product units a computationally powerful and biologically plausible extension to backpropagation networks we introduce a new form of computational unit for feedforward learning networks of the backpropagation type instead of calculating a weighted sum this unit calculates a weighted product where each input is raised to a power determined by a variable weight such a unit can learn an arbitrary polynomial term which would then feed into higher level standard summing units we show how learning operates with product units provide examples to show their efficiency for various types of problems and argue that they naturally extend the family of theoretical feedforward net structures there is a plausible neurobiological interpretation for one interesting configuration of product and summing units

approximation capabilities of multilayer fuzzy neural networks on the set of fuzzy valued functions this paper first introduces a piecewise linear interpolation method for fuzzy valued functions based on this we present a concrete approximation procedure to show the capability of four layer regular fuzzy neural networks to perform approximation on the set of all d p continuous fuzzy valued functions this approach can also be used to approximate d continuous fuzzy valued functions an example is given to illustrate the approximation procedure

direct adaptive control of partially known nonlinear systems a direct adaptive control strategy for a class of single input single output nonlinear systems is presented the major advantage of the proposed method is that a detailed dynamic nonlinear model is not required for controller design the only information required about the plant is measurements of the state variables the relative degree and the sign of a lie derivative which appears in the associated input output linearizing control law unknown controller functions are approximated using locally supported radial basis functions that are introduced only in regions of the state space where the closed loop system actually evolves lyapunov stability analysis is used to derive parameter update laws which ensure under certain assumptions the state vector remains bounded and the plant output asymptotically tracks the output of a linear reference model the technique is successfully applied to a nonlinear biochemical reactor model

on repeated root cyclic codes a parity check matrix for a q ary repeated root cyclic code is derived using the hasse derivative then the minimum distance of a q ary repeated root cyclic code is expressed in terms of the minimum distance of a certain simple root cyclic code with the help of this result several binary repeated root cyclic codes of lengths up to n 62 are shown to contain the largest known number of codewords for their given length and minimum distance the relative minimum distance d sub min n of q ary repeated root cyclic codes of rate r or r is proven to tend to zero as the largest multiplicity of a root of the generator g x increases to infinity it is further shown that repeated root cycle codes cannot be asymptotically better than simple root cyclic codes

analysis of augmented input layer rbfnn in this paper we present and analyze a new structure for designing a radial basis function neural network rbfnn in the training phase input layer of rbfnn is augmented with desired output vector generalization phase involves the following steps 1 identify the cluster to which a previously unseen input vector belongs 2 augment the input layer with an average of the targets of the input vectors in the identified cluster and 3 use the augmented network to estimate the unknown target it is shown that under some reasonable assumptions the generalization error function admits an upper bound in terms of the quantization errors minimized when determining the centers of the proposed method over the training set and the difference between training samples and generalization samples in a deterministic setting when the difference between the training and generalization samples goes to zero the upper bound can be made arbitrarily small by increasing the number of hidden neurons computer simulations verified the effectiveness of the proposed method

a polynomial time algorithm for generating neural networks for pattern classification its stability properties and some test results polynomial time training and network design are two major issues for the neural network community a new algorithm has been developed that can learn in polynomial time and also design an appropriate network the algorithm is for classification problems and uses linear programing models to design and train the network this paper summarizes the new algorithm proves its stability properties and provides some computational results to demonstrate its potential

back to the future radial basis function network revisited radial basis function rbf networks are a classical family of algorithms for supervised learning the most popular approach for training rbf networks has relied on kernel methods using regularization based on a norm in a reproducing kernel hilbert space rkhs which is a principled and empirically successful framework in this paper we aim to revisit some of the older approaches to training the rbf networks from a more modern perspective specifically we analyze two common regularization procedures one based on the square norm of the coefficients in the network and another one using centers obtained by k k means clustering we show that both of these rbf methods can be recast as certain data dependent kernels we provide a theoretical analysis of these methods as well as a number of experimental results pointing out very competitive experimental performance as well as certain advantages over the standard kernel methods in terms of both flexibility incorporating of unlabeled data and computational complexity finally our results shed light on some impressive recent successes of using soft k k means features for image recognition and other tasks

solving a system of nonlinear integral equations by an rbf network in this paper a novel learning strategy for radial basis function networks rbfn is proposed by adjusting the parameters of the hidden layer including the rbf centers and widths the weights of the output layer are adapted by local optimization methods a new local optimization algorithm based on a combination of the gradient and newton methods is introduced the efficiency of some local optimization methods to update the weights of rbfn is studied in solving systems of nonlinear integral equations

precision and approximate flatness in artificial neural networks several of the major classes of artificial neural networks output functions are linear combinations of elements of approximately flat sets this gives a tool for understanding the precision problem as well as providing a rationale for mixing types of networks approximate flatness also helps explain the power of artificial neural network techniques relative to series regressions series regressions take linear combinations of flat sets while neural networks take linear combinations of the much larger class of approximately flat sets

on the geometry of feedforward neural network error surfaces many feedforward neural network architectures have the property that their overall input output function is unchanged by certain weight permutations and sign flips in this paper the geometric structure of these equioutput weight space transformations is explored for the case of multilayer perceptron networks with tanh activation functions similar results hold for many other types of neural networks it is shown that these transformations form an algebraic group isomorphic to a direct product of weyl groups results concerning the root spaces of the lie algebras associated with these weyl groups are then used to derive sets of simple equations for minimal sufficient search sets in weight space these sets which take the geometric forms of a wedge and a cone occupy only a minute fraction of the volume of weight space a separate analysis shows that large numbers of copies of a network performance function optimum weight vector are created by the action of the equioutput transformation group and that these copies all lie on the same sphere some implications of these results for learning are discussed

a quasi armax approach to modelling of non linear systems this paper proposes a class of quasi armax models for non linear systems similar to ordinary non linear armax models the quasi armax models are flexible black box models but they have various linearity properties similar to those of linear armax models a modelling scheme is introduced to construct models consisting of two parts a macro part and a kernel part by using taylor expansion and other mathematical transformation techniques it is first constructed as a class of quasi armax interfaces macro parts that have various linearity properties but contain some complicated coefficients mimo neurofuzzy models kernel parts are then introduced to represent the complicated coefficients it is shown that the proposed quasi armax models have both good approximation ability and some easy to use properties the proposed models have been successfully applied to prediction fault detection and adaptive control of non linear systems

reformulated radial basis neural networks trained by gradient descent this paper presents an axiomatic approach for constructing radial basis function rbf neural networks this approach results in a broad variety of admissible rbf models including those employing gaussian rbfs the form of the rbfs is determined by a generator function new rbf models can be developed according to the proposed approach by selecting generator functions other than exponential ones which lead to gaussian rbfs this paper also proposes a supervised learning algorithm based on gradient descent for training reformulated rbf neural networks constructed using the proposed approach a sensitivity analysis of the proposed algorithm relates the properties of rbfs with the convergence of gradient descent learning experiments involving a variety of reformulated rbf networks generated by linear and exponential generator functions indicate that gradient descent learning is simple easily implementable and produces rbf networks that perform considerably better than conventional rbf models trained by existing algorithms

efficient relighting of rbf based illumination adjustable images an illumination adjustable image iai contains a large number of prerecorded images under various light directions relighting a scene under complicated lighting conditions can be achieved from the iai using the radial basis function rbf approach to represent an iai is proven to be more efficient than using the spherical harmonic approach however to represent high frequency lighting effects we need to use many rbfs hence the relighting speed could be very slow this brief investigates a partial reconstruction scheme for relighting an iai based on the locality of rbfs compared with the conventional rbf and spherical harmonics sh approaches the proposed scheme has a much faster relighting speed under the similar distortion performance

an experimental study on nonlinear function computation for neural fuzzy hardware design an experimental study on the influence of the computation of basic nodal nonlinear functions on the performance of nfss is described in this paper systems architecture size their approximation capability and the smoothness of provided mappings are used as performance indexes for this comparative paper two widely used kernel functions the sigmoid logistic function and the gaussian function are analyzed by their computation through an accuracy controllable approximation algorithm designed for hardware implementation two artificial neural network ann paradigms are selected for the analysis backpropagation neural networks bpnns with one hidden layer and radial basis function rbf networks extensive simulation of simple benchmark approximation problems is used in order to achieve generalizable conclusions for the performance analysis of fuzzy systems a functional equivalence theorem is used to extend obtained results to fuzzy inference systems fiss finally the adaptive neurofuzzy inference system anfis paradigm is used to observe the behavior of neurofuzzy systems with learning capabilities

indexed families of functionals and gaussian radial basis functions we report on results concerning the capabilities of gaussian radial basis function networks in the setting of inner product spaces that need not be finite dimensional specifically we show that important indexed families of functionals can be uniformly approximated with the approximation uniform also with respect to the index applications are described concerning the classification of signals and the synthesis of reconfigurable classifiers

multiobjective evolutionary optimization of the size shape and position parameters of radial basis function networks for function approximation this paper presents a multiobjective evolutionary algorithm to optimize radial basis function neural networks rbfnns in order to approach target functions from a set of input output pairs the procedure allows the application of heuristics to improve the solution of the problem at hand by including some new genetic operators in the evolutionary process these new operators are based on two well known matrix transformations singular value decomposition svd and orthogonal least squares ols which have been used to define new mutation operators that produce local or global modifications in the radial basis functions rbfs of the networks the individuals in the population in the evolutionary procedure after analyzing the efficiency of the different operators we have shown that the global mutation operators yield an improved procedure to adjust the parameters of the rbfnns

approximation and radial basis function networks this paper concerns conditions for the approximation of functions in certain general spaces using radial basis function networks it has been shown in recent papers that certain classes of radial basis function networks are broad enough for universal approximation in this paper these results are considerably extended and sharpened

approximating gaussian mixture model or radial basis function network with multilayer perceptron gaussian mixture models gmms and multilayer perceptron mlp are both popular pattern classification techniques this brief shows that a multilayer perceptron with quadratic inputs mlpq can accurately approximate gmms with diagonal covariance matrices the mapping equations between the parameters of gmm and the weights of mlpq are presented a similar approach is applied to radial basis function networks rbfns to show that rbfns with gaussian basis functions and euclidean norm can be approximated accurately with mlpq the mapping equations between rbfn and mlpq weights are presented there are well established training procedures for gmms such as the expectation maximization em algorithm the gmm parameters obtained by the em algorithm can be used to generate a set of initial weights of mlpq similarly a trained rbfn can be used to generate a set of initial weights of mlpq mlpq training can be continued further with gradient descent based methods which can lead to improvement in performance compared to the gmm or rbfn from which it is initialized thus the mlpq can always perform as well as or better than the gmm or rbfn

statistical theory of learning curves under entropic loss criterion the present paper elucidates a universal property of learning curves which shows how the generalization error training error and the complexity of the underlying stochastic machine are related and how the behavior of a stochastic machine is improved as the number of training examples increases the error is measured by the entropic loss it is proved that the generalization error converges to h0 the entropy of the conditional distribution of the true machine as h0 m 2t while the training error converges as h0 m 2t where t is the number of examples and m shows the complexity of the network when the model is faithful implying that the true machine is in the model m is reduced to m the number of modifiable parameters this is a universal law because it holds for any regular machine irrespective of its structure under the maximum likelihood estimator similar relations are obtained for the bayes and gibbs learning algorithms these learning curves show the relation among the accuracy of learning the complexity of a model and the number of training examples

time varying radial basis functions we introduce radial basis functions rbfs whose time varying coefficients determine not only the amplitude and position of each rbf but also their shape the intended use of these time varying rbfs tv rbfs is in the local in time representation of low dimensional approximations of functions that arise in solving spatiotemporal evolution problems in particular for time varying spatially localized solutions with a temporal translation component such as traveling waves modulated pulses or soliton like solutions of evolutionary differential equations this paper is restricted to the one dimensional spatial case we also present an algorithm that places the time varying rbfs tv rbfs over spatiotemporal data that may come from experiments from finely discretized pde simulations or even from multiscale particle based simulations it first approximates the function at a single time instant a temporal snapshot as a sum of rbfs using a novel weighted minimization that causes each rbf to primarily approximate one of the localized parts of the function it then extends that approximation to tv rbfs over a sequence of snapshots of the function at different times we conclude by discussing the potential uses of these tv rbfs

efficient training of neural nets for nonlinear adaptive filtering using a recursive levenberg marquardt algorithm the levenberg marquardt algorithm is often superior to other training algorithms in off line applications this motivates the proposal of using a recursive version of the algorithm for on line training of neural nets for nonlinear adaptive filtering the performance of the suggested algorithm is compared with other alternative recursive algorithms such as the recursive version of the off line steepest descent and gauss newton algorithms the advantages and disadvantages of the different algorithms are pointed out the algorithms are tested on some examples and it is shown that generally the recursive levenberg marquardt algorithm has better convergence properties than the other algorithms

comparison of some neural network and scattered data approximations the inverse manipulator kinematics example this paper compares the application of five different methods for the approximation of the inverse kinematics of a manipulator arm from a number of joint angle cartesian coordinate training pairs the first method is a standard feedforward neural network with error backpropagation learning the next two methods are derived from an extended kohonen map algorithm that we combine with shepard interpolation for the forward computation we compare the method of ritter et al for the learning of the extended kohonen map to our own scheme based on gradient descent optimization we also study three scattered data approximation algorithms they include two variants of the radial basis function rbf method hardy s multiquadrics and gaussian rbf we further develop our own local polynomial fit method that could be considered as a modification of mclain s method we propose extensions to the considered scattered data approximation algorithms to make them suitable for vector valued multivariable functions such as the mapping of cartesian coordinates into joint angle coordinates

an involutory pascal matrix abstract an involutory upper triangular pascal matrix u n is investigated eigenvectors of u n and of u n t are considered a characterization of u n is obtained and it is shown how the results can be extended to matrices over a commutative ring with unity

a neural network learning theory and a polynomial time rbf algorithm this paper presents a new learning theory a set of principles for brain like learning and a corresponding algorithm for the neural network field the learning theory defines computational characteristics that are much more brain like than that of classical connectionist learning robust and reliable learning algorithms would result if these learning principles are followed rigorously when developing neural network algorithms this paper also presents a new algorithm for generating radial basis function rbf nets for function approximation the design of the algorithm is based on the proposed set of learning principles the net generated by this algorithm is not a typical rbf net but a combination of truncated rbf and other types of hidden units the algorithm uses random clustering and linear programming lp to design and train this mixed rbf net polynomial time complexity of the algorithm is proven and computational results are provided for the well known mackey glass chaotic time series problem the logistic map prediction problem various neuro control problems and several time series forecasting problems the algorithm can also be implemented as an online adaptive algorithm

discrete time model reference adaptive control of nonlinear dynamical systems using neural networks a multilayer discrete time neural net nn controller is presented for the direct model reference adaptive control of a class of multi input multi output mim0 nonlinear dynamical systems the nonlinear dynamical system is assumed to be controllable and its state vector is available for measurement the nn controller exhibits learning while functioning features instead of learning then control so that control is immediate with no explicit learning phase needed the tracking error between the output of a nonlinear plant and an ideal linear model converges within a very short time persistence of excitation pe is not needed linearity in the parameters is not required and certainty equivalence is not used this overcomes several limitations of standard adaptive control the novel weight tuning paradigm for the nn controller is based on the well known delta rule but includes a modification to the learning rate parameter plus a correction term it guarantees tracking as well as bounded nn weights in non i

repeated root cyclic codes in the theory of cyclic codes it is common practice to require that n q 1 where n is the word length and f sub q is the alphabet it is shown that the even weight subcodes of the shortened binary hamming codes form a sequence of repeated root cyclic codes that are optimal in nearly all other cases one does not find good cyclic codes by dropping the usual restriction that n and q must be relatively prime this statement is based on an analysis for lengths up to 100 a theorem shows why this was to be expected but it also leads to low complexity decoding methods this is an advantage especially for the codes that are not much worse than corresponding codes of odd length it is demonstrated that a binary cyclic code of length 2n n odd can be obtained from two cyclic codes of length n by the well known mod u mod u v mod construction this leads to an infinite sequence of optimal cyclic codes with distance 4 furthermore it is shown that low complexity decoding methods can be used for these codes the structure theorem generalizes to other characteristics and to other lengths some comparisons of the methods using earlier examples are given

some identities on catalan numbers and hypergeometric functions via catalan matrix power abstract in this paper we use the catalan matrix power as a tool for deriving identities involving catalan numbers and hypergeometric functions for that purpose we extend earlier investigated relations between the catalan matrix and the pascal matrix by inserting the catalan matrix power and particulary the squared catalan matrix in those relations we also pay attention to some relations between catalan matrix powers of different degrees which allows us to derive the simplification formula for hypergeometric function 3 f 2 as well as the simplification formula for the product of the catalan number and the hypergeometric function 3 f 2 some identities involving catalan numbers proved by the non matrix approach are also given

fuzzy systems as universal approximators an additive fuzzy system can uniformly approximate any real continuous function on a compact domain to any degree of accuracy an additive fuzzy system approximates the function by covering its graph with fuzzy patches in the input output state space and averaging patches that overlap the fuzzy system computes a conditional expectation e y x if we view the fuzzy sets as random sets each fuzzy rule defines a fuzzy patch and connects commonsense knowledge with state space geometry neural or statistical clustering systems can approximate the unknown fuzzy patches from training data these adaptive fuzzy systems approximate a function at two levels at the local level the neural system approximates and tunes the fuzzy rules at the global level the rules or patches approximate the function

on simultaneous approximations by radial basis function neural networks it is shown in this paper by a constructive method that any multivariate function and all its existing derivatives can be simultaneously approximated by a radial basis function rbf neural network where the assumptions on the rbfs are relatively mild by considering the translations of rbfs it is proved that an o m 1 2 order of simultaneous approximations to a certain class of functions can be achieved by using a translation network with just m neurons

wavelet neural networks for function learning a wavelet based neural network is described the structure of this network is similar to that of the radial basis function rbf network except that in the present paper the radial basis functions are replaced by orthonormal scaling functions that are not necessarily radial symmetric the efficacy of this type of network in function learning and estimation is demonstrated through theoretical analysis and experimental results in particular it has been shown that the wavelet network has universal and l sup 2 approximation properties and is a consistent function estimator convergence rates associated with these properties are obtained for certain function classes where the rates avoid the curse of dimensionality in the experiments the wavelet network performed well and compared favorably to the mlp and rbf networks

bch codes for the rosenbloom tsfasman metric the rosenbloom tsfasman metric has attracted the attention of many researchers as a generalization of the hamming metric that is relevant to practical problems codes for this metric were considered in particular reed solomon codes were generalized to be compatible with this metric in this paper a generalization of bch codes for the rosenbloom tsfasman metric is proposed this generalization is based on considering bch codes as subfield subcodes of reed solomon codes by characterizing these subfield subcodes an explicit construction of bch codes for the rosenbloom tsfasman metric is provided two important properties of reed solomon codes and bch codes for the rosenbloom tsfasman metric are studied and compared with those for the hamming metric these properties are cyclic structure and duality the approach is based on galois fourier transforms associated with hasse derivatives

object classification in 3 d images using alpha trimmed mean radial basis function network we propose a pattern classification based approach for simultaneous three dimensional 3 d object modeling and segmentation in image volumes the 3 d objects are described as a set of overlapping ellipsoids the segmentation relies on the geometrical model and graylevel statistics the characteristic parameters of the ellipsoids and of the graylevel statistics are embedded in a radial basis function rbf network and they are found by means of unsupervised training a new robust training algorithm for rbf networks based on spl alpha trimmed mean statistics is employed in this study the extension of the hough transform algorithm in the 3 d space by employing a spherical coordinate system is used for ellipsoidal center estimation we study the performance of the proposed algorithm and we present results when segmenting a stack of microscopy images

modeling multivariate time series on manifolds with skew radial basis functions we present an approach for constructing nonlinear empirical mappings from high dimensional domains to multivariate ranges we employ radial basis functions and skew radial basis functions for constructing a model using data that are potentially scattered or sparse the algorithm progresses iteratively adding a new function at each step to refine the model the placement of the functions is driven by a statistical hypothesis test that accounts for correlation in the multivariate range variables the test is applied on training and validation data and reveals nonstatistical or geometric structure when it fails at each step the added function is fit to data contained in a spatiotemporally defined local region to determine the parameters in particular the scale of the local model the scale of the function is determined by the zero crossings of the autocorrelation function of the residuals the model parameters and the number of basis functions are determined automatically from the given data and there is no need to initialize any ad hoc parameters save for the selection of the skew radial basis functions compactly supported skew radial basis functions are employed to improve model accuracy order and convergence properties the extension of the algorithm to higher dimensional ranges produces reduced order models by exploiting the existence of correlation in the range variable data structure is tested not just in a single time series but between all pairs of time series we illustrate the new methodologies using several illustrative problems including modeling data on manifolds and the prediction of chaotic time series

multiwavelet neural network and its approximation properties a model of multiwavelet based neural networks is proposed its universal and l sup 2 approximation properties together with its consistency are proved and the convergence rates associated with these properties are estimated the structure of this network is similar to that of the wavelet network except that the orthonormal scaling functions are replaced by orthonormal multiscaling functions the theoretical analyses show that the multiwavelet network converges more rapidly than the wavelet network especially for smooth functions to make a comparison between both networks experiments are carried out with the lemarie meyer wavelet network the daubechies2 wavelet network and the ghm multiwavelet network and the results support the theoretical analysis well in addition the results also illustrate that at the jump discontinuities the approximation performance of the two networks are about the same

embedding dimension estimation of chaotic time series using self generating radial basis function network abstract in this paper we apply the self generating radial basis function network sgrbf to the dimension analysis of the nonlinear dynamical systems including chaotic time series firstly we formulate a nonlinear time series identification problem with a nonlinear autoregressive moving average narmax model secondly we propose an identification algorithm using sgrbf which is regarded as both a three layer network or a fuzzy model of class c with gaussian membership function we apply this method to the estimation of embedding dimension for chaotic time series since the embedding dimension plays an essential role for the identification and the prediction of nonlinear dynamical systems including chaos in this estimation method identification problems with gradually increasing embedding dimension are solved and the identified result is used for computing correlation coefficients between the predicted time series and the observed one we apply this method to the embedding dimension estimation of a henon map and a chaotic pulsation time series in a finger s capillary vessels

sup norm approximation bounds for networks through probabilistic methods we consider the problem of approximating a smooth target function and its derivatives by networks involving superpositions and translations of a fixed activation function the approximation is with respect to the sup norm and the rate is shown to be of order o n sup 1 2 that is the rate is independent of the dimension d the results apply to neural and wavelet networks and extend the work of barren see proc 7th yale workshop on adaptive and learning systems may 1992 and ibid vol 39 p 930 1993 the approach involves probabilistic methods based on central limit theorems for empirical processes indexed by classes of functions

approximately universal codes over slow fading channels performance of reliable communication over a coherent slow fading multiple input multiple output mimo channel at high signal to noise ratio snr is succinctly captured as a fundamental tradeoff between diversity and multiplexing gains this paper studies the problem of designing codes that optimally tradeoff the diversity and multiplexing gains the main contribution is a precise characterization of codes that are universally tradeoff optimal i e they optimally tradeoff the diversity and multiplexing gains for every statistical characterization of the fading channel this characterization is referred to as approximate universality the approximation is in the connection between error probability and outage capacity with diversity and multiplexing gains respectively the characterization of approximate universality is then used to construct new coding schemes as well as to show optimality of several schemes proposed in the space time coding literature

activation functions defined on higher dimensional spaces for approximation on compact sets with and without scaling let g be a slowly increasing function of locally bounded variation defined on rc 1 c d we investigate when g can be an activation function of the hidden layer units of three layer neural networks that approximate continuous functions on compact sets if the support of the fourier transform of g includes a converging sequence of points with distinct distances from the origin it can be an activation function without scaling if and only if the support of its fourier transform includes a point other than the origin it can be an activation function with scaling we also look for a condition on which an activation function can be used for approximation without rotation any nonpolynomial functions can be activation functions with scaling and many familiar functions such as sigmoid functions and radial basis functions can be activation functions without scaling with or without scaling some of them defined on rd can be used without rotation even if they are not spherically symmetric

descartes rule of signs for radial basis function neural networks we establish versions of descartes rule of signs for radial basis function rbf neural networks the rbf rules of signs provide tight bounds for the number of zeros of univariate networks with certain parameter restrictions moreover they can be used to infer that the vapnik chervonenkis vc dimension and pseudodimension of these networks are no more than linear this contrasts with previous work showing that rbf neural networks with two or more input nodes have superlinear vc dimension the rules also give rise to lower bounds for network sizes thus demonstrating the relevance of network parameters for the complexity of computing with rbf neural networks

control of chaotic dynamical systems using radial basis function network approximators abstract this paper presents a general control method based on radial basis function networks rbfns for chaotic dynamical systems for many chaotic systems that can be decomposed into a sum of a linear and a nonlinear part under some mild conditions the rbfn can be used to well approximate the nonlinear part of the system dynamics the resulting system is then dominated by the linear part with some small or weak residual nonlinearities due to the rbfn approximation errors thus a simple linear state feedback controller can be devised to drive the system response to a desirable set point in addition to some theoretical analysis computer simulations on two representative continuous time chaotic systems the duffing and the lorenz systems are presented to demonstrate the effectiveness of the proposed method

universal approximation using radial basis function networks there have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables some of these studies deal with cases in which the hidden layer nonlinearity is not a sigmoid this was motivated by successful applications of feedforward networks with nonsigmoidal hidden layer units this paper reports on a related study of radial basis function rbf networks and it is proved that rbf networks having one hidden layer are capable of universal approximation here the emphasis is on the case of typical rbf networks and the results show that a certain class of rbf networks with the same smoothing factor in each kernel node is broad enough for universal approximation

a recursive local polynomial approximation method using dirichlet clouds and radial basis functions we present a recursive function approximation technique that does not require the storage of the arrival data stream our work is motivated by algorithms in stochastic optimization which require approximating functions in a recursive setting such as a stochastic approximation algorithm the unique collection of these features in this technique is essential for nonlinear modeling of large data sets where the storage of the data becomes prohibitively expensive and in circumstances where our knowledge about a given query point increases as new information arrives the algorithm presented here employs radial basis functions rbfs to provide locally adaptive parametric models such as linear models the local models are updated using recursive least squares and only store the statistical representative of the local approximations the resulting scheme is very fast and memory efficient without compromising accuracy in comparison to methods well accepted as the standard and some advanced techniques used for fu

stirling matrix via pascal matrix abstract the pascal type matrices obtained from the stirling numbers of the first kind s n k and of the second kind s n k are studied respectively it is shown that these matrices can be factorized by the pascal matrices also the ldu factorization of a vandermonde matrix of the form v n x x 1 x n 1 for any real number x is obtained furthermore some well known combinatorial identities are obtained from the matrix representation of the stirling numbers and these matrices are generalized in one or two variables

efficient algorithm for training interpolation rbf networks with equally spaced nodes this brief paper proposes a new algorithm to train interpolation gaussian radial basis function rbf networks in order to solve the problem of interpolating multivariate functions with equally spaced nodes based on an efficient two phase algorithm recently proposed by the authors euclidean norm associated to gaussian rbf is now replaced by a conveniently chosen mahalanobis norm that allows for directly computing the width parameters of gaussian radial basis functions the weighting parameters are then determined by a simple iterative method the original two phase algorithm becomes a one phase one simulation results show that the generality of networks trained by this new algorithm is sensibly improved and the running time significantly reduced especially when the number of nodes is large

a general fuzzy scheme for nonlinear processes with stability analysis abstract in this paper we develop a general fuzzy control scheme for nonlinear processes assuming little knowledge about the dynamics of the controlled process the proposed scheme starts by probing the process at different points in its operating region to generate a fuzzy quantisation a simple local controller is then designed at each fuzzy locality a fuzzy inference mechanism then links up tje local controllers to form a global controller which can be further refined by the learning algorithm by employing a newly developed structure adaptive fuzzy modelling scheme the appropriate fuzzy rule base for the inference mechanism can be extracted stably and efficiently the conditions for the stability of the global controller are rigourously established simulation results are presented to illustrate the effectiveness of the scheme

nonparametric estimation and classification using radial basis function nets and empirical risk minimization studies convergence properties of radial basis function rbf networks for a large class of basis functions and reviews the methods and results related to this topic the authors obtain the network parameters through empirical risk minimization the authors show the optimal nets to be consistent in the problem of nonlinear function approximation and in nonparametric classification for the classification problem the authors consider two approaches the selection of the rbf classifier via nonlinear function estimation and the direct method of minimizing the empirical error probability the tools used in the analysis include distribution free nonasymptotic probability inequalities and covering numbers for classes of functions

neural networks with local receptive fields and superlinear vc dimension local receptive field neurons comprise such well known and widely used unit types as radial basis function rbf neurons and neurons with center surround receptive field we study the vapnik chervonenkis vc dimension of feedforward neural networks with one hidden layer of these units for several variants of local receptive field neurons we show that the vc dimension of these networks is superlinear in particular we establish the bound w log k for any reasonably sized network with w parameters and k hidden nodes this bound is shown to hold for discrete center surround receptive field neurons which are physiologically relevant models of cells in the mammalian visual system for neurons computing a difference of gaussians which are popular in computational vision and for standard rbf neurons a major alternative to sigmoidal neurons in artificial neural networks the resutlt for rbf neural networks is of particular interest since it answers a question that has been open for several years the results also give rise to lower bounds for networks with fixed input dimension regarding constants all bounds are larger than those known thus far for similar architectures with sigmoidal neurons the superlinear lower bounds contrast with linear upper bounds for single local receptive field neurons also derived here

predicting the future advantages of semilocal units in investigating gaussian radial basis function rbf networks for their ability to model nonlinear time series we have found that while rbf networks are much faster than standard sigmoid unit backpropagation for low dimensional problems their advantages diminish in high dimensional input spaces this is particularly troublesome if the input space contains irrelevant variables we suggest that this limitation is due to the localized nature of rbfs to gain the advantages of the highly nonlocal sigmoids and the speed advantages of rbfs we propose a particular class of semilocal activation functions that is a natural interpolation between these two families we present evidence that networks using these gaussian bar units avoid the slow learning problem of sigmoid unit networks and very importantly are more accurate than rbf networks in the presence of irrelevant inputs on the mackey glass and coupled lattice map problems the speedup over sigmoid networks is so dramatic that the difference in training time between rbf and gaussian bar networks is minor gaussian bar architectures that superpose composed gaussians gaussians of gaussians to approximate the unknown function have the best performance we postulate that an interesing behavior displayed by gaussian bar functions under gradient descent dynamics which we call automatic connection pruning is an important factor in the success of this representation

median radial basis function neural network radial basis functions rbfs consist of a two layer neural network where each hidden unit implements a kernel function each kernel is associated with an activation region from the input space and its output is fed to an output unit in order to find the parameters of a neural network which embeds this structure we take into consideration two different statistical approaches the first approach uses classical estimation in the learning stage and it is based on the learning vector quantization algorithm and its second order statistics extension after the presentation of this approach we introduce the median radial basis function mrbf algorithm based on robust estimation of the hidden unit parameters the proposed algorithm employs the marginal median for kernel location estimation and the median of the absolute deviations for the scale parameter estimation a histogram based fast implementation is provided for the mrbf algorithm the theoretical performance of the two training algorithms is comparatively evaluated when estimating the network weights the network is applied in pattern classification problems and in optical flow segmentation

layered neural networks with gaussian hidden units as universal approximations a neural network with a single layer of hidden units of gaussian type is proved to be a universal approximator for real valued maps defined on convex compact sets of rn

mixture of experts regression modeling by deterministic annealing we propose a new learning algorithm for regression modeling the method is especially suitable for optimizing neural network structures that are amenable to a statistical description as mixture models these include mixture of experts hierarchical mixture of experts hme and normalized radial basis functions nrbf unlike recent maximum likelihood ml approaches we directly minimize the squared regression error we use the probabilistic framework as means to define an optimization method that avoids many shallow local minima on the complex cost surface our method is based on deterministic annealing da where the entropy of the system is gradually reduced with the expected regression cost energy minimized at each entropy level the corresponding lagrangian is the system s free energy and this annealing process is controlled by variation of the lagrange multiplier which acts as a temperature parameter the new method consistently and substantially outperformed the competing methods for training nrbf and hme regression functions over a variety of benchmark regression examples

a unified approach for the catalan matrices by using riordan arrays abstract we study generalized catalan matrices based on the riordan array and fuss catalan numbers a unified combinatorial interpretation for the entries of the generalized catalan matrices is presented by means of m dyck paths some properties and examples of the generalized catalan matrices are given including a new convolution formula for the generalized catalan numbers finally we present applications of generalized catalan matrices to the problems in counting the hill free and lower peak free m dyck paths

adaptive control of a class of nonlinear systems with fuzzy logic an adaptive tracking control architecture is proposed for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture employs fuzzy systems which are expressed as a series expansion of basis functions to adaptively compensate for the plant nonlinearities global asymptotic stability of the algorithm is established in the lyapunov sense with tracking errors converging to a neighborhood of zero simulation results for an unstable nonlinear plant are included to demonstrate that incorporating the linguistic fuzzy information from human experts results in superior tracking performance

orthogonal least squares learning algorithm for radial basis function networks the radial basis function network offers a viable alternative to the two layer neural network in many applications of signal processing a common learning algorithm for radial basis function networks is based on first choosing randomly some data points as radial basis function centers and then using singular value decomposition to solve for the weights of the network such a procedure has several drawbacks and in particular an arbitrary selection of centers is clearly unsatisfactory the authors propose an alternative learning procedure based on the orthogonal least squares method the procedure chooses radial basis function centers one by one in a rational way until an adequate network has been constructed in the algorithm each selected center maximizes the increment to the explained variance or energy of the desired output and does not suffer numerical ill conditioning problems the orthogonal least squares learning strategy provides a simple and efficient means for fitting radial basis function networks this is illustrated using examples taken from two different signal processing applications

higher order statistics based radial basis function networks for signal enhancement in this paper a higher order statistics hos based radial basis function rbf network for signal enhancement is introduced in the proposed scheme higher order cumulants of the reference signal were used as the input of hos based rbf an hos based supervised learning algorithm with mean square error obtained from higher order cumulants of the desired input and the system output as the learning criterion was used to adapt weights the motivation is that the hos can effectively suppress gaussian and symmetrically distributed non gaussian noise the influence of a gaussian noise on the input of hos based rbf and the hos based learning algorithm can be mitigated simulated results indicate that hos based rbf can provide better performance for signal enhancement under different noise levels and its performance is insensitive to the selection of learning rates moreover the efficiency of hos based rbf under the nonstationary gaussian noise is stable

on the existence of universally decodable matrices universally decodable matrices udms can be used for coding purposes when transmitting over slow fading channels these matrices are parameterized by positive integers l and n and a prime power q the main result of this correspondence is that the simple condition l q 1 is both necessary and sufficient for l n q vdms to exist the existence proof is constructive and yields a coding scheme that is equivalent to a class of codes that was proposed by rosenbloom and tsfasman our work resolves an open problem posed recently in the literature

catalan matrix and related combinatorial identities we introduce the notion of the catalan matrix c n x whose non zero elements are expressions which contain the catalan numbers arranged into a lower triangular toeplitz matrix inverse of the catalan matrix is derived correlations between the matrix c n x and the generalized pascal matrix are considered some combinatorial identities involving catalan numbers binomial coefficients and the generalized hypergeometric function are derived using these correlations moreover an additional explicit representation of the catalan number as well as an explicit representation of the sum of the first m catalan numbers are given

function approximation fast convergence neural approach based on spectral analysis we propose a constructive approach to building single hidden layer neural networks for nonlinear function approximation using frequency domain analysis we introduce a spectrum based learning procedure that minimizes the difference between the spectrum of the training data and the spectrum of the network s estimates the network is built up incrementally during training and automatically determines the appropriate number of hidden units this technique achieves similar or better approximation with faster convergence times than traditional techniques such as backpropagation

maximum likelihood neural approximation in presence of additive colored noise in many practical situations the noise samples may be correlated in this case the estimation of noise parameters can be used to improve the approximation estimation of the noise structure can also be used to find a stopping criterion in constructive neural networks to avoid overfitting a network construction procedure must be stopped when residual can be considered as noise the knowledge on the noise may be used for whitening the residual so that a correlation hypothesis test determines if the network growing must be continued or not in this paper supposing a gaussian noise model we study the problem of multi output nonlinear regression using mlp when the noise in each output is a correlated autoregressive time series and is spatially correlated with other output noises we show that the noise parameters can be determined simultaneously with the network weights and used to construct an estimator with a smaller variance and so to improve the network generalization performance moreover if a constructive procedure is used to build the network the estimated parameters may be used to stop the procedure

growing radial basis neural networks merging supervised and unsupervised learning with network growth techniques this paper proposes a framework for constructing and training radial basis function rbf neural networks the proposed growing radial basis function grbf network begins with a small number of prototypes which determine the locations of radial basis functions in the process of training the grbf network gross by splitting one of the prototypes at each growing cycle two splitting criteria are proposed to determine which prototype to split in each growing cycle the proposed hybrid learning scheme provides a framework for incorporating existing algorithms in the training of grbf networks these include unsupervised algorithms for clustering and learning vector quantization as well as learning algorithms for training single layer linear neural networks a supervised learning scheme based on the minimization of the localized class conditional variance is also proposed and tested grbf neural networks are evaluated and tested on a variety of data sets with very satisfactory results

approximation in weighted p mean by rbf networks of delsarte translates abstract we give conditions on the kernel function or activation function for the family of radial basis function rbf neural networks obtained upon replacing the usual translation by the delsarte one with not necessarily the same smoothing factor in all kernel nodes to have the universal approximation property in suitable weighted l p spaces 1 p a complete characterization of such kernels for p 1 and p 2 is provided

a radial basis function algorithm with automatic model order determination we present a new radial basis function rbf algorithm for constructing nonlinear models from data that may be sparsely scattered in high dimensions we propose a simplified method for identifying function locations that is based on an approximation to the autocorrelation function local regions are now defined based on zero crossings of the modified autocorrelation contribution the resulting local spatiotemporal data appear to capture increased structure as evidenced by the reduced number of basis functions required to achieve algorithm convergence and the fact that the resulting approximations have significantly improved accuracy we prove a zero crossing lemma and convergence of the algorithm in the norm for continuous functions over a compact domain further we show that several hypotheses tests can be used to detect this convergence the model parameters and the number of basis functions are determined automatically from the given data the only user parameter is the confidence level of the hypothe

preliminary study on wilcoxon learning machines as is well known in statistics the resulting linear regressors by using the rank based wilcoxon approach to linear regression problems are usually robust against or insensitive to outliers this motivates us to introduce in this paper the wilcoxon approach to the area of machine learning specifically we investigate four new learning machines namely wilcoxon neural network wnn wilcoxon generalized radial basis function network wgrbfn wilcoxon fuzzy neural network wfnn and kernel based wilcoxon regressor kwr these provide alternative learning machines when faced with general nonlinear learning problems simple weights updating rules based on gradient descent will be derived some numerical examples will be provided to compare the robustness against outliers for various learning machines simulation results show that the wilcoxon learning machines proposed in this paper have good robustness against outliers we firmly believe that the wilcoxon approach will provide a promising methodology for many machine learning problems

robust h control of multiple time delay uncertain nonlinear system using fuzzy model and adaptive neural network in this paper a robust control method which combines fuzzy model based control with adaptive neural network control is presented for a class of uncertain nonlinear system with multiple time delay the fuzzy t s model with multiple time delay is adopted for the approximate modeling of the nonlinear system with some unknown uncertainties and fuzzy model based h control law is designed by means of lmi method a full adaptive rbf neural network is added to the fuzzy h control in order to guarantee the robust stability of the controlled system the effect of the unknown uncertainties and the error caused by fuzzy modeling can be overcome by adaptive tuning of the weights centers and widths of the rbf neural network on line and no constraint or matching conditions are required the stability of the designed closed loop system is thus proved the proposed method is applied to a multiple time delay nonlinear chaotic system and the simulation results show that the proposed method cannot only stabilize the chaos systems but has strong robustness against uncertainties and external disturbance

an rbf based compression method for image based relighting in image based relighting a pixel is associated with a number of sampled radiance values this paper presents a two level compression method in the first level the plenoptic property of a pixel is approximated by a spherical radial basis function srbf network that means that the spherical plenoptic function of each pixel is represented by a number of srbf weights in the second level we apply a wavelet based method to compress these srbf weights to reduce the visual artifact due to quantization noise we develop a constrained method for estimating the srbf weights our proposed approach is superior to jpeg jpeg2000 and mpeg compared with the spherical harmonics approach our approach has a lower complexity while the visual quality is comparable the real time rendering method for our srbf representation is also discussed

optical flow estimation and moving object segmentation based on median radial basis function network various approaches have been proposed for simultaneous optical flow estimation and segmentation in image sequences in this study the moving scene is decomposed into different regions with respect to their motion by means of a pattern recognition scheme the inputs of the proposed scheme are the feature vectors representing still image and motion information each class corresponds to a moving object the classifier employed is the median radial basis function mrbf neural network an error criterion function derived from the probability estimation theory and expressed as a function of the moving scene model is used as the cost function each basis function is activated by a certain image region marginal median and median of the absolute deviations from the median mad estimators are employed for estimating the basis function parameters the image regions associated with the basis functions are merged by the output units in order to identify moving objects

improved data visualisation through nonlinear dissimilarity modelling inherent to state of the art dimension reduction algorithms is the assumption that global distances between observations are euclidean despite the potential for altogether non euclidean data manifolds we demonstrate that a non euclidean manifold chart can be approximated by implementing a universal approximator over a dictionary of dissimilarity measures building on recent developments in the field this approach is transferable across domains such that observations can be vectors distributions graphs and time series for instance our novel dissimilarity learning method is illustrated with four standard visualisation datasets showing the benefits over the linear dissimilarity learning approach

weighted gradient enhanced kriging for high dimensional surrogate modeling and design optimization a novel formulation of gradient enhanced surrogate model called weighted gradient enhanced kriging is proposed and used in combination with the cheap gradients obtained by the adjoint method to a

a hybrid constructive algorithm for single layer feedforward networks learning single layer feedforward networks slfns have been proven to be a universal approximator when all the parameters are allowed to be adjustable it is widely used in classification and regression problems the slfn learning involves two tasks determining network size and training the parameters most current algorithms could not be satisfactory to both sides some algorithms focused on construction and only tuned part of the parameters which may not be able to achieve a compact network other gradient based optimization algorithms focused on parameters tuning while the network size has to be preset by the user therefore trial and error approach has to be used to search the optimal network size because results of each trial cannot be reused in another trial it costs much computation in this paper a hybrid constructive hc algorithm is proposed for slfn learning which can train all the parameters and determine the network size simultaneously at first by combining levenberg marquardt algorithm and least square method a hybrid algorithm is presented for training slfn with fixed network size then with the hybrid algorithm an incremental constructive scheme is proposed a new randomly initialized neuron is added each time when the training entrapped into local minima because the training continued on previous results after adding new neurons the proposed hc algorithm works efficiently several practical problems were given for comparison with other popular algorithms the experimental results demonstrated that the hc algorithm worked more efficiently than those optimization methods with trial and error and could achieve much more compact slfn than those construction algorithms

on global local artificial neural networks for function approximation we present a hybrid radial basis function rbf sigmoid neural network with a three step training algorithm that utilizes both global search and gradient descent training the algorithm used is intended to identify global features of an input output relationship before adding local detail to the approximating function it aims to achieve efficient function approximation through the separate identification of aspects of a relationship that are expressed universally from those that vary only within particular regions of the input space we test the effectiveness of our method using five regression tasks four use synthetic datasets while the last problem uses real world data on the wave overtopping of seawalls it is shown that the hybrid architecture is often superior to architectures containing neurons of a single type in several ways lower mean square errors are often achievable using fewer hidden neurons and with less need for regularization our global local artificial neural network gl ann is also seen to compare favorably with both perceptron radial basis net and regression tree derived rbfs a number of issues concerning the training of gl anns are discussed the use of regularization the inclusion of a gradient descent optimization step the choice of rbf spreads model selection and the development of appropriate stopping criteria

parameter estimation of sigmoid superpositions dynamical system approach superposition of sigmoid function over a finite time interval is shown to be equivalent to the linear combination of the solutions of a linearly parameterized system of logistic differential equations due to the linearity with respect to the parameters of the system it is possible to design an effective procedure for parameter adjustment stability properties of this procedure are analyzed

approximation with random bases in this work we discuss the problem of selecting suitable approximators from families of parameterized elementary functions that are known to be dense in a hilbert space of functions we consider and analyze published procedures both randomized and deterministic for selecting elements from these families that have been shown to ensure the rate of convergence in l2 norm of order o 1 n where n is the number of elements we show that both randomized and deterministic procedures are successful if additional information about the families of functions to be approximated is provided in the absence of such additional information one may observe exponential growth of the number of terms needed to approximate the function and or extreme sensitivity of the outcome of the approximation to parameters implications of our analysis for applications of neural networks in modeling and control are illustrated with examples

a comparison of the von mises and gaussian basis functions for approximating spherical acoustic scatter this paper compares the approximation accuracy of two basis functions that share a common radial basis function rbf neural network architecture used for approximating a known function on the unit sphere the basis function types considered are that of a new spherical basis function the von mises function and the now well known gaussian basis function gradient descent learning rules were applied to optimize learn the solution for both approximating basis functions a benchmark approximation problem was used to compare the performance of the two types of basis functions in this case the mathematical expression for the scattering of an acoustic wave striking a rigid sphere

on functional approximation with normalized gaussian units feedforward neural networks with a single hidden layer using normalized gaussian units are studied it is proved that such neural networks are capable of universal approximation in a satisfactory sense then a hybrid learning rule as per moody and darken that combines unsupervised learning of hidden units and supervised learning of output units is considered by using the method of ordinary differential equations for adaptive algorithms ode method it is shown that the asymptotic properties of the learning rule may be studied in terms of an autonomous cascade of dynamical systems some recent results from hirsch about cascades are used to show the asymptotic stability of the learning rule