on the inequivalence of generalized preparata codes if m is odd and sigma in aut gf 2 m is such that x rightarrow x sigma 2 1 is 1 1 there is a 2 m 1 1 2 m l 2m 2 nonlinear binary code p sigma having minimum distance 5 all the codes p sigma have the same distance and weight enumerators as the usual preparata codes which rise as p sigma when x sigma x 2 it is shown that p sigma and p tau are equivalent if and only if tau sigma pm 1 and aut p sigma is determined

second order volterra filtering and its application to nonlinear system identification some recent results on the design and implementation of second order volterra filters are presented the second order volterra filter is a nonlinear filter with the filter structure of second order volterra series a simple minimum mean square error solution for the volterra filter is derived based on the assumption that the filter input is gaussian also we propose an iterative factorization technique to design a subclass of the volterra filters which can alleviate the complexity of the filtering operations considerably furthermore an adaptive algorithm for the volterra filter is investigated along with its mean convergence and asymptotic excess mean square error finally the utility of the volterra filter is demonstrated by utilizing it in studies of nonlinear drift oscillations of moored vessels subject to random sea waves

cocyclic hadamard matrices from forms over finite frobenius rings abstract we give a construction of cocyclic butson hadamard matrices from bilinear forms over finite frobenius rings which generalizes previous constructions we then prove a classification result showing that the resulting matrices depend only on the additive group of the underlying module of the form

fuzzy granular gravitational clustering algorithm for multivariate data abstract a new method for finding fuzzy information granules from multivariate data through a gravitational inspired clustering algorithm is proposed in this paper the proposed algorithm incorporates the theory of granular computing which adapts the cluster size with respect to the context of the given data via an inspiration in newton s law of universal gravitation both conditions of clustering similar data and adapting to the size of each granule are achieved this paper compares the fuzzy granular gravitational clustering algorithm fggca against other clustering techniques on two grounds classification accuracy and clustering validity indices e g rand fm davies bouldin dunn homogeneity and separation the fggca is tested with multiple benchmark classification datasets such as iris wine seeds and glass identification

nonlinear digital filter realization by distributed arithmetic it is assumed that a nonlinear shift invariant digital filter is represented by a truncated discrete volterra series a matrix notation based on the properties of the kronecker product of matrices is derived for the nonlinear operator of order k in the volterra expansion the matrix notation shows how the well known distributed arithmetic originally proposed for linear digital filters can be adopted for the realization of nonlinear operators the characteristics of combinatorial and memory oriented structures are compared and the problem of complexity reduction is discussed

manifold regularization a geometric framework for learning from labeled and unlabeled examples we propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution we focus on a semi supervised framework that incorporates labeled and unlabeled data in a general purpose learner some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases we use properties of reproducing kernel hilbert spaces to prove new representer theorems that provide theoretical basis for the algorithms as a result in contrast to purely graph based approaches we obtain a natural out of sample extension to novel examples and so are able to handle both transductive and truly semi supervised settings we present experimental evidence suggesting that our semi supervised algorithms are able to use unlabeled data effectively finally we have a brief discussion of unsupervised and fully supervised learning within our general framework

a variable step adaptation algorithm for memory oriented volterra filters a recently proposed adaptation algorithm for digital non linear adaptive filters based on the truncated discrete volterra series is modified and improved in order to get a faster and more accurate convergence performance examples are presented

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

optimizing the kernel in the empirical feature space in this paper we present a method of kernel optimization by maximizing a measure of class separability in the empirical feature space an euclidean space in which the training data are embedded in such a way that the geometrical structure of the data in the feature space is preserved employing a data dependent kernel we derive an effective kernel optimization algorithm that maximizes the class separability of the data in the empirical feature space it is shown that there exists a close relationship between the class separability measure introduced here and the alignment measure defined recently by cristianini extensive simulations are carried out which show that the optimized kernel is more adaptive to the input data and leads to a substantial sometimes significant improvement in the performance of various data classification algorithms

binary sequences with gold like correlation but larger linear span a new construction of optimal binary sequences identical to the well known family of gold sequences in terms of maximum nontrivial correlation magnitude and family size but having larger linear span is presented the distribution of correlation values is determined for every odd integer spl tau spl ges 3 the construction provides a family that contains 2 sup spl tau 1 cyclically distinct sequences each of period 2 sup spl tau 1 the maximum nontrivial correlation magnitude equals 2 spl tau 1 sup 2 1 with one exception each of the sequences in the family has linear span at least spl tau sup 2 spl tau 2 compared to 2 spl tau for gold sequences the sequences are easily implemented using a quaternary shift register followed by a simple feedforward nonlinearity

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

4 phase sequences with near optimum correlation properties two families of four phase sequences are constructed using irreducible polynomials over z sub 4 family a has period l 2 sup r 1 size l 2 and maximum nontrivial correlation magnitude c sub max

orbit and coset analysis of the golay and related codes let b be a code of length n over a field f with automorphism group g b sub w denotes the subset of codewords of weight w the goal is to classify the vectors of f sup n into orbits under g and to determine their distances from the various subcodes b sub w this is done for the first order reed muller nordstrom robinson and hamming codes of length 16 the golay and shortened golay codes of lengths 22 23 24 and the ternary golay code of length 12

translates of linear codes over z sub 4 we give a method to compute the complete weight distribution of translates of linear codes over z sub 4 the method follows known ideas that have already been used successfully by others for hamming weight distributions for the particular case of quaternary preparata codes we obtain that the number of distinct complete weights for the dual preparata codes and the number of distinct complete coset weight enumerators for the preparata codes are both equal to ten independent of the code length

a novel radial basis function neural network for discriminant analysis a novel radial basis function neural network for discriminant analysis is presented in this paper in contrast to many other researches this work focuses on the exploitation of the weight structure of radial basis function neural networks using the bayesian method it is expected that the performance of a radial basis function neural network with a well explored weight structure can be improved as the weight structure of a radial basis function neural network is commonly unknown the bayesian method is therefore used in this paper to study this a priori structure two weight structures are investigated in this study i e a single gaussian structure and a two gaussian structure an expectation maximization learning algorithm is used to estimate the weights the simulation results showed that the proposed radial basis function neural network with a weight structure of two gaussians outperformed the other algorithms

simultaneous clustering and classification over cluster structure representation two main tasks in pattern recognition area are clustering and classification owing to their different goals traditionally these two tasks are treated separately however when label information is available such separate treatment can not fully explore data information first classification is not favored by the data cluster structure second clustering is not guided by valuable label information third the relationship of clusters and classes is not revealed contrary to this separate learning treatment simultaneous learning clustering and classification could benefit each other and overcomes these problems recently a simultaneous learning framework scc was proposed through modeling p class cluster classification and clustering mechanism in scc depend only on cluster centroids however it produces severely nonlinear objective thus has to use a heuristic searching method modified particle swarm optimization to find the optimal solution but it is very slow further modeling p class cluster makes scc hard to incorporate semi supervised settings in this paper we propose an alternative framework sc 3sr for simultaneous learning besides a classifier derived on the original data another classifier on the newly formed cluster structure representation is derived as well through this classifier the clustering learning is guided by the label and classification learning is also favored by cluster structure of data the final objective is continuously differentiable for which some principled optimization algorithms with convergence guaranteed exist as a result our algorithm is much faster than scc further we generalize this framework to semisupervised situation with the idea of manifold regularization and propose semisc 3sr algorithm our experiments demonstrate the effectiveness of both sc 3sr and semisc 3sr

a simultaneous learning framework for clustering and classification traditional pattern recognition generally involves two tasks unsupervised clustering and supervised classification when class information is available fusing the advantages of both clustering learning and classification learning into a single framework is an important problem worthy of study to date most algorithms generally treat clustering learning and classification learning in a sequential or two step manner i e first execute clustering learning to explore structures in data and then perform classification learning on top of the obtained structural information however such sequential algorithms cannot always guarantee the simultaneous optimality for both clustering and classification learning in fact the clustering learning in these algorithms just aids the subsequent classification learning and does not benefit from the latter to overcome this problem a simultaneous learning framework for clustering and classification scc is presented in this paper scc aims to achieve three goals 1 acquiring the robust classification and clustering simultaneously 2 designing an effective and transparent classification mechanism 3 revealing the underlying relationship between clusters and classes to this end with the bayesian theory and the cluster posterior probabilities of classes we define a single objective function to which the clustering process is directly embedded by optimizing this objective function the effective and robust clustering and classification results are achieved simultaneously experimental results on both synthetic and real life datasets show that scc achieves promising classification and clustering results at one time

covering radius survey and recent results all known results on covering radius are presented as well as some new results there are a number of upper and lower bounds including asymptotic results a few exact determinations of covering radius some extensive relations with other aspects of coding theory through the reed muller codes and new results on the least covering radius of any linear n k code there is also a recent result on the complexity of computing the covering radius

a linear construction for certain kerdock and preparata codes the nordstrom robinson kerdock and slightly modified preparata codes are shown to be linear over z4 the integers mod 4 the kerdock and preparata codes are duals over z4 and the nordstrom robinson code is self dual all these codes are just extended cyclic codes over z4 this provides a simple definition for these codes and explains why their hamming weight distributions are dual to each other firstand second order reed muller codes are also linear codes over z4 but hamming codes in general are not nor is the golay code

fast ml decoding algorithm for the nordstrom robinson code corresp the extended nordstrom robinson code is an optimum nonlinear double error correcting code of length 16 with considerable practical importance this code is also useful as a rate 1 2 vector quantizer for random waveforms such as speech linear predictive coding lpc residual a fast decoding algorithm is described for maximum likelihood decoding or nearest neighbor search in the squared error sense with 304 additions and 128 comparisons

an exponential number of generalized kerdock codes if n 1 is an odd composite integer then there are at least 2 1 2 n pairwise inequivalent binary error correcting codes of length 2 n size 2 2 n and minimum distance 2 n 1 2 1 2 n 1

enhancing prototype reduction schemes with lvq3 type algorithms abstract various prototype reduction schemes have been reported in the literature foremost among these are the prototypes for nearest neighbor pnn the vector quantization vq and the support vector machines svm methods in this paper we shall show that these schemes can be enhanced by the introduction of a post processing phase that is related but not identical to the lvq3 process although the post processing with lvq3 has been reported for the som and the basic vq methods in this paper we shall show that an analogous philosophy can be used in conjunction with the svm and pnn rules our essential modification to lvq3 first entails a partitioning of the respective training sets into two sets called the placement set and the optimizing set which are instrumental in determining the lvq3 parameters such a partitioning is novel to the literature our experimental results demonstrate that the proposed enhancement yields the best reported prototype condensation scheme to date for both artificial data sets and for samples involving real life data sets

cocyclic simplex codes of type spl alpha over z sub 4 and z sub 2 sup s over the past decade cocycles have been used to construct hadamard and generalized hadamard matrices this in turn has led to the construction of codes self dual and others here we explore these ideas further to construct cocyclic complex and butson hadamard matrices and subsequently we use the matrices to construct simplex codes of type spl alpha over z 4 and z 2 sup s respectively

input space versus feature space in kernel based methods this paper collects some ideas targeted at advancing our understanding of the feature spaces associated with support vector sv kernel functions we first discuss the geometry of feature space in particular we review what is known about the shape of the image of input space under the feature space map and how this influences the capacity of sv methods following this we describe how the metric governing the intrinsic geometry of the mapped surface can be computed in terms of the kernel using the example of the class of inhomogeneous polynomial kernels which are often used in sv pattern recognition we then discuss the connection between feature space and input space by dealing with the question of how one can given some vector in feature space find a preimage exact or approximate in input space we describe algorithms to tackle this issue and show their utility in two applications of kernel methods first we use it to reduce the computational complexity of sv decision functions second we combine it with the kernel pca algorithm thereby constructing a nonlinear statistical denoising technique which is shown to perform well on real world data

discriminatively regularized least squares classification over the past decades regularization theory is widely applied in various areas of machine learning to derive a large family of novel algorithms traditionally regularization focuses on smoothing only and does not fully utilize the underlying discriminative knowledge which is vital for classification in this paper we propose a novel regularization algorithm in the least squares sense called discriminatively regularized least squares classification drlsc method which is specifically designed for classification inspired by several new geometrically motivated methods drlsc directly embeds the discriminative information as well as the local geometry of the samples into the regularization term so that it can explore as much underlying knowledge inside the samples as possible and aim to maximize the margins between the samples of different classes in each local area furthermore by embedding equality type constraints in the formulation the solutions of drlsc can follow from solving a set of linear equations and the framework naturally contains multi class problems experiments on both toy and real world problems demonstrate that drlsc is often superior in classification performance to the classical regularization algorithms including regularization networks support vector machines and some of the recent studied manifold regularization techniques