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

monotonicity method applied to the complex ginzburg landau and related equations abstract global existence of unique strong solutions is established for the complex ginzburg landau equation tu i u i u p 1u u 0 where 0 0 r p 1 and 1 2 p p 1 the key is a new inequality in monotonicity methods it is based on the sectorial estimates of in lp 1 and the nonlinear operator u u p 1u appearing in the equation the key inequality also yields the global existence of unique strong solutions of the nonlinear schrodinger type equation with monotone nonlinearity tu i u u p 1u 0 for all p 1

soft learning vector quantization and clustering algorithms based on non euclidean norms single norm algorithms this paper presents the development of soft clustering and learning vector quantization lvq algorithms that rely on a weighted norm to measure the distance between the feature vectors and their prototypes the development of lvq and clustering algorithms is based on the minimization of a reformulation function under the constraint that the generalized mean of the norm weights be constant according to the proposed formulation the norm weights can be computed from the data in an iterative fashion together with the prototypes an error analysis provides some guidelines for selecting the parameter involved in the definition of the generalized mean in terms of the feature variances the algorithms produced from this formulation are easy to implement and they are almost as fast as clustering algorithms relying on the euclidean norm an experimental evaluation on four data sets indicates that the proposed algorithms outperform consistently clustering algorithms relying on the euclidean norm and they are strong competitors to non euclidean algorithms which are computationally more demanding

image compression based on fuzzy algorithms for learning vector quantization and wavelet image decomposition this paper evaluates the performance of an image compression system based on wavelet based subband decomposition and vector quantization the images are decomposed using wavelet filters into a set of subbands with different resolutions corresponding to different frequency bands the resulting subbands are vector quantized using the linde buzo gray 1980 algorithm and various fuzzy algorithms for learning vector quantization falvq these algorithms perform vector quantization by updating all prototypes of a competitive neural network through an unsupervised learning process the quality of the multiresolution codebooks designed by these algorithms is measured on the reconstructed images belonging to the training set used for multiresolution codebook design and the reconstructed images from a testing set

switching regression models and fuzzy clustering a family of objective functions called fuzzy c regression models which can be used too fit switching regression models to certain types of mixed data is presented minimization of particular objective functions in the family yields simultaneous estimates for the parameters of c regression models together with a fuzzy c partitioning of the data a general optimization approach for the family of objective functions is given and corresponding theoretical convergence results are discussed the approach is illustrated by two numerical examples that show how it can be used to fit mixed data to coupled linear and nonlinear models

existence and decay estimates of solutions to complex ginzburg landau type equations abstract this paper deals with the initial boundary value problem denoted by cgl for the complex ginzburg landau type equation u t i u i u q 1 u u 0 with initial data u 0 l p in the case 1 q 1 2 p n where is bounded or unbounded in r n 0 r there are a lot of studies on local and global existence of solutions to cgl including the physically relevant case q 3 and 0 this paper gives existence results with precise properties of solutions and rigorous proof from a mathematical point of view the physically relevant case can be considered as a special case of the results moreover in the case 0 local and global existence of solutions with the decay estimate u t l p c 1 e c 2 t c 1 c 2 are positive constants is obtained under some conditions the key to the local existence is to construct a semigroup e t i and its l p l q estimate on the other hand the key to the global existence is to derive estimates for solutions by using a kind of interpolation inequality with re v p 2 v i v

fuzzy vector quantization algorithms and their application in image compression this paper presents the development and evaluation of fuzzy vector quantization algorithms these algorithms are designed to achieve the quality of vector quantizers provided by sophisticated but computationally demanding approaches while capturing the advantages of the frequently used in practice k means algorithm such as speed simplicity and conceptual appeal the uncertainty typically associated with clustering tasks is formulated in this approach by allowing the assignment of each training vector to multiple clusters in the early stages of the iterative codebook design process a training vector assignment strategy is also proposed for the transition from the fuzzy mode where each training vector can be assigned to multiple clusters to the crisp mode where each training vector can be assigned to only one cluster such a strategy reduces the dependence of the resulting codebook on the random initial codebook selection the resulting algorithms are used in image compression based on vector quantization this application provides the basis for evaluating the computational efficiency of the proposed algorithms and comparing the quality of the resulting codebook design with that provided by competing techniques

an axiomatic approach to soft learning vector quantization and clustering this paper presents an axiomatic approach to soft learning vector quantization lvq and clustering based on reformulation the reformulation of the fuzzy c means fcm algorithm provides the basis for reformulating entropy constrained fuzzy clustering ecfc algorithms according to the proposed approach the development of specific algorithms reduces to the selection of a generator function linear generator functions lead to the fcm and fuzzy learning vector quantization algorithms while exponential generator functions lead to ecfc and entropy constrained learning vector quantization algorithms the reformulation of lvq and clustering algorithms also provides the basis for developing uncertainty measures that can identify feature vectors equidistant from all prototypes these measures are employed by a procedure developed to make soft lvq and clustering algorithms capable of identifying outliers in the data set this procedure is evaluated by testing the algorithms generated by linear and exponential generator functions on speech data

rough fuzzy set based image compression a new coding decoding scheme based on the properties and operations of rough fuzzy sets is presented by normalizing pixel values of an image each pixel value can be interpreted as the degree of belonging of that pixel to the image foreground the image is then subdivided into blocks which are partitioned and characterized by a pair of approximation sets coding uses a codebook created with a quantization algorithm to find the best approximating pair for each block while decoding exploits specific properties of rough fuzzy sets to rebuild the blocks the method called by us rough fuzzy vector quantization rfvq relies on the representation capabilities of the vector to be quantized and not on the quantization algorithm to determine optimal codevectors a comparison with other fuzzy based coding decoding schemes and with dct and jpeg methods is performed by means of peak signal to noise ratio psnr values results show that for low compression rates the proposed method performs well and in some cases the psnr obtained with rfvq is close to the jpeg s psnr

well posedness and energy decay estimates in the cauchy problem for the damped defocusing schrodinger equation abstract in this paper we study the existence at the h 1 level as well as the stability for the damped defocusing schrodinger equation in r d the considered damping coefficient is time dependent and may vanish at infinity to prove the existence we employ the method devised by ozsari kalantarov and lasiecka 27 which is based on monotone operators theory in particular when d 1 or d 2 we obtain the uniqueness decay estimates for the l 2 level and h 1 l p 2 level energies are established with the help of direct multipliers method coupled with refined energy estimates and a lower semi continuity argument

global attractors for the complex ginzburg landau equation abstract this paper concerns the long time behavior of the following complex ginzburg landau equations u t i u i u p 2 u u 0 without any restriction on p 2 under the assumptions 1 4 we first prove the well posedness of strong solutions for the complex ginzburg landau equations and then the existence of absorbing sets in l 2 h 0 1 l p and h 2 l 2 p 1 respectively for the semigroup s t t 0 generated by 1 1 1 3 is established finally we prove the existence of global attractors in l 2 and h 0 1 for the semigroup s t t 0 generated by 1 1 1 3 by the sobolev compactness embedding theorem and prove the existence of global attractor in l p for the semigroup s t t 0 generated by 1 1 1 3 using interpolation inequality

a methodology for constructing fuzzy algorithms for learning vector quantization this paper presents a general methodology for the development of fuzzy algorithms for learning vector quantization falvq the design of specific falvq algorithms according to existing approaches reduces to the selection of the membership function assigned to the weight vectors of an lvq competitive neural network which represent the prototypes the development of a broad variety of falvq algorithms can be accomplished by selecting the form of the interference function that determines the effect of the nonwinning prototypes on the attraction between the winning prototype and the input of the network the proposed methodology provides the basis for extending the existing falvq 1 falvq 2 and falvq 3 families of algorithms this paper also introduces two quantitative measures which establish a relationship between the formulation that led to falvq algorithms and the competition between the prototypes during the learning process the proposed algorithms and competition measures are tested and evaluated using the iris data set the significance of the proposed competition measure is illustrated using falvq algorithms to perform segmentation of magnetic resonance images of the brain

semigroups of locally lipschitz operators associated with semilinear evolution equations of parabolic type abstract a characterization problem is discussed of semigroups of locally lipschitz operators providing mild solutions to the cauchy problem for the semilinear evolution equation of parabolic type u t a b u t for t 0 by parabolic type we mean that the operator a is the infinitesimal generator of an analytic c 0 semigroup on a general banach space x the operator b is assumed to be locally continuous from a subset of y into x where y is a banach space which is contained in x and has a stronger norm defined through a fractional power of a the characterization is applied to the global solvability of the mixed problem for the complex ginzburg landau equation

an interactive multiple objective linear programming method for a class of underlying nonlinear utility functions this paper develops a method for interactive multiple objective linear programming assuming an unknown pseudo concave utility function satisfying certain general properties the method is an extension of our earlier method published in this journal zionts s wallenius j 1976 an interactive programming method for solving the multiple criteria problem management sci 22 6 652 663 various technical problems present in predecessor versions have been resolved in addition to presenting the supporting theory and algorithm we discuss certain options in implementation and summarize our practical experience with several versions of the method

fuzzy algorithms for learning vector quantization this paper presents the development of fuzzy algorithms for learning vector quantization falvq these algorithms are derived by minimizing the weighted sum of the squared euclidean distances between an input vector which represents a feature vector and the weight vectors of a competitive learning vector quantization lvq network which represent the prototypes this formulation leads to competitive algorithms which allow each input vector to attract all prototypes the strength of attraction between each input and the prototypes is determined by a set of membership functions which can be selected on the basis of specific criteria a gradient descent based learning rule is derived for a general class of admissible membership functions which satisfy certain properties the falvq 1 falvq 2 and falvq 3 families of algorithms are developed by selecting admissible membership functions with different properties the proposed algorithms are tested and evaluated using the iris data set the efficiency of the proposed algorithms is also illustrated by their use in codebook design required for image compression based on vector quantization

an integrated approach to fuzzy learning vector quantization and fuzzy c means clustering derives an interpretation for a family of competitive learning algorithms and investigates their relationship to fuzzy c means and fuzzy learning vector quantization these algorithms map a set of feature vectors into a set of prototypes associated with a competitive network that performs unsupervised learning derivation of the new algorithms is accomplished by minimizing an average generalized distance between the feature vectors and prototypes using gradient descent a close relationship between the resulting algorithms and fuzzy c means is revealed by investigating the functionals involved it is also shown that the fuzzy c means and fuzzy learning vector quantization algorithms are related to the proposed algorithms if the learning rate at each iteration is selected to satisfy a certain condition

a fuzzy vector quantization approach to image compression the use of fuzzy clustering analysis in the early stages of a vector quantization process is able to make this process less sensitive to initialization this is justified by the fact that fuzzy clustering provides a framework for the quantitative formulation of the uncertainty typically involved in a training vector space this paper proposes a fuzzy clustering based vector quantization algorithm which employs an effective vector assignment strategy for the transition from fuzzy mode where each training vector is assigned to more than one clusters to crisp mode where each training vector is assigned to only one cluster this transition is controlled by analytical conditions that are obtained by minimizing a modified objective function for the fuzzy c means algorithm the application to image compression shows that the proposed approach is able to achieve a very efficient performance while maintaining the computational capabilities of other methods reported in the literature

soft learning vector quantization and clustering algorithms based on ordered weighted aggregation operators this paper presents the development and investigates the properties of ordered weighted learning vector quantization lvq and clustering algorithms these algorithms are developed by using gradient descent to minimize reformulation functions based on aggregation operators an axiomatic approach provides conditions for selecting aggregation operators that lead to admissible reformulation functions minimization of admissible reformulation functions based on ordered weighted aggregation operators produces a family of soft lvq and clustering algorithms which includes fuzzy lvq and clustering algorithms as special cases the proposed lvq and clustering algorithms are used to perform segmentation of magnetic resonance mr images of the brain the diagnostic value of the segmented mr images provides the basis for evaluating a variety of ordered weighted lvq and clustering algorithms

repairs to glvq a new family of competitive learning schemes first we identify an algorithmic defect of the generalized learning vector quantization glvq scheme that causes it to behave erratically for a certain scaling of the input data we show that glvq can behave incorrectly because its learning rates are reciprocally dependent on the sum of squares of distances from an input vector to the node weight vectors finally we propose a new family of models the glvq f family that remedies the problem we derive competitive learning algorithms for each member of the glvq f model and prove that they are invariant to all scalings of the data we show that glvq f offers a wide range of learning models since it reduces to lvq as its weighting exponent a parameter of the algorithm approaches one from above as this parameter increases glvq f then transitions to a model in which either all nodes may be excited according to their inverse distances from an input or in which the winner is excited while losers are penalized and as this parameter increases without limit glvq f updates all nodes equally we illustrate the failure of glvq and success of glvq f with the iris data

proper efficiency in nonconvex multicriteria programming proper efficient solutions of nonconvex vector maximum problems can be generated by solving a parametric family of ordinary nonlinear programs this parametric scheme follows from the characterization of proper efficiency by an extended form of the generalized tchebycheff norm

optimization of clustering criteria by reformulation various hard fuzzy and possibilistic clustering criteria objective functions are useful as bases for a variety of pattern recognition problems at present many of these criteria have customized individual optimization algorithms because of the specialized nature of these algorithms experimentation with new and existing criteria can be very inconvenient and costly in terms of development and implementation time this paper shows how to reformulate some clustering criteria so that specialized algorithms can be replaced by general optimization routines found in commercially available software we prove that the original and reformulated versions of each criterion are fully equivalent finally two numerical examples are given to illustrate reformulation

a heuristic for estimating nadir criterion values in multiple objective linear programming in this paper we further investigate the problem of finding nadir criterion values minimum criterion values over the nondominated set in multiple objective linear programming although easy to obtain the minimum values present in a payoff table are unreliable and should only be used with caution especially in problems that have more than a small number of extreme points to obtain better estimates of the nadir criterion values without adding great complexity to the task we present an approach based upon the use of reference directions at each iteration of this approach a reference direction is chosen that maximally minimizes the objective under consideration we proceed with reference directions that accomplish this until the objective under consideration reaches a local minimum over the nondominated set then a cutting plane is inserted into the problem and another direction if one can be found that maximally minimizes the objective under consideration is employed although the method is heurist

dynamic parametric bounds on efficient outcomes in interactive multiple criteria decision making problems abstract bounds on efficient outcomes in interactive multiple criteria decision making problems are derived bounds are dynamic i e they become stronger with the growing number of explicitly identified outcomes they are also parametric with respect to weighting coefficients computational cost to calculate bounds is negligible bounds of the sort offer a breakthrough for prohibitive size and or solution time bottlenecks by allowing a decision maker to interact with an approximation of the underlying mathematical model rather the model itself possible applications of bounds to existing interactive decision making algorithms are discussed illustrative numerical examples are given

segmentation of magnetic resonance images using fuzzy algorithms for learning vector quantization evaluates a segmentation technique for magnetic resonance mr images of the brain based on fuzzy algorithms for learning vector quantization falvq these algorithms perform vector quantization by updating all prototypes of a competitive network through an unsupervised learning process segmentation of mr images is formulated as an unsupervised vector quantization process where the local values of different relaxation parameters form the feature vectors which are represented by a relatively small set of prototypes some experiments are presented which evaluate a variety of falvq algorithms in terms of their ability to identify different tissues and discriminate between normal tissues and abnormalities

a simple approach to the cauchy problem for complex ginzburg landau equations by compactness methods abstract this paper is concerned with the cauchy problem cgl in l 2 r n for complex ginzburg landau equations with laplacian and nonlinear term u q 2 u multiplied by the complex coefficients i and i respectively q 2 0 0 r the global existence of strong solutions to cgl is established without any upper restriction on q 2 but with some restriction on and the result corresponds to ginibre and velo 1996 3 proposition 5 1 which is technically proved by combining convolution regularizing methods with compactness localizing methods while our proof here is fairly simplified the key to our proof is the cauchy problem cgl r which is cgl with replaced with v r where v r x x r 2 x r v r x 0 x r the solvability of cgl r is a direct consequence of okazawa and yokota 2002 16 theorem 4 1 taking the limit of global strong solutions to cgl r as r yields a global strong solution to cgl the result gives also an unbounded version of okazawa and yokota 2002 16 theorem 1 1 with p 2 for the initial boundary value problem on bounded domains

the cauchy problem in local spaces for the complex ginzburg landau equation i compactness methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

the cauchy problem in local spaces for the complex ginzburg landau equation ii contraction methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces