quantization methods for equal gain transmission with finite rate feedback we consider the design and analysis of quantizers for equal gain transmission egt systems with finite rate feedback based communication in flat fading multiple input single output miso systems egt is a beamforming technique that maximizes the miso channel capacity when there is an equal power per antenna constraint at the transmitter and requires the feedback of t 1 phase angles when there are t antennas at the transmitter in this paper we contrast two popular approaches for quantizing the phase angles vector quantization vq and scalar quantization sq on the vq side using the capacity loss with respect to egt with perfect channel information at transmitter as performance metric we develop a criterion for designing the beamforming codebook for quantized egt q egt we also propose an iterative algorithm based on the well known generalized lloyd algorithm for computing the beamforming vector codebook on the analytical side we study the performance of q egt and derive closed form expressions for the performance in terms of capacity loss and outage probability in the case of i i d rayleigh flat fading channels on the sq side assuming uniform scalar quantization and i i d rayleigh flat fading channels we derive the high resolution performance of quantized egt and contrast the performance with that of vq we find that although both vq and sq achieve the same rate of convergence to the capacity with perfect feedback as the number of feedback bits b increases there exists a fixed gap between the two

invertible preservers and algebraic groups abstract we consider groups of invertible linear transformations on matrices preserving some property let f be an infinite field our main result is that if an invertible transformation of m n f preserves a finite union of similarity classes then the transformation is essentially conjugation this was proved for f algebraically closed of characteristic zero by li and pierce our method relies heavily on results about maximal subgroups of algebraic groups these results were obtained by dynkin in characteristic zero and seitz in positive characteristic

high rate vector quantization for detection we investigate high rate quantization for various detection and reconstruction loss criteria a new distortion measure is introduced which accounts for global loss in best attainable binary hypothesis testing performance the distortion criterion is related to the area under the receiver operating characteristic roc curve specifically motivated by sanov s theorem we define a performance curve as the trajectory of the pair of optimal asymptotic type i and type ii error rates of the most powerful neyman pearson test of the hypotheses the distortion measure is then defined as the difference between the area under the curve auc of the optimal pre encoded hypothesis test and the auc of the optimal post encoded hypothesis test as compared to many previously introduced distortion measures for decision making this distortion measure has the advantage of being independent of any detection thresholds or priors on the hypotheses which are generally difficult to specify in the code design process a high resolution zador gersho type of analysis is applied to characterize the point density and the inertial profile associated with the optimal high rate vector quantizer the analysis applies to a restricted class of high rate quantizers that have bounded cells with vanishing volumes the optimal point density is used to specify a lloyd type algorithm which allocates its finest resolution to regions where the gradient of the pre encoded likelihood ratio has greatest magnitude

high resolution quantization theory and the vector quantizer advantage the authors consider how much performance advantage a fixed dimensional vector quantizer can gain over a scalar quantizer they collect several results from high resolution or asymptotic in rate quantization theory and use them to identify source and system characteristics that contribute to the vector quantizer advantage one well known advantage is due to improvement in the space filling properties of polytopes as the dimension increases others depend on the source s memory and marginal density shape the advantages are used to gain insight into product transform lattice predictive pyramid and universal quantizers although numerical prediction consistently overestimated gains in low rate 1 bit sample experiments the theoretical insights may be useful even at these rates

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

finding the number of clusters in a dataset one of the most difficult problems in cluster analysis is identifying the number of groups in a dataset most previously suggested approaches to this problem are either somewhat ad hoc or require parametric assumptions and complicated calculations in this article we develop a simple yet powerful nonparametric method for choosing the number of clusters based on distortion a quantity that measures the average distance per dimension between each observation and its closest cluster center our technique is computationally efficient and straightforward to implement we demonstrate empirically its effectiveness not only for choosing the number of clusters but also for identifying underlying structure on a wide range of simulated and real world datasets in addition we give a rigorous theoretical justification for the method based on information theoretic ideas specifically results from the subfield of electrical engineering known as rate distortion theory allow us to describe the behavior of the dist

high rate vector quantization for the neyman pearson detection of correlated processes this paper investigates the effect of quantization on the performance of the neyman pearson test it is assumed that a sensing unit observes samples of a correlated stationary ergodic multivariate process each sample is passed through an n point quantizer and transmitted to a decision device which performs a binary hypothesis test for any false alarm level it is shown that the miss probability of the neyman pearson test converges to zero exponentially as the number of samples tends to infinity assuming that the observed process satisfies certain mixing conditions the main contribution of this paper is to provide a compact closed form expression of the error exponent in the high rate regime i e when the number n of quantization levels tends to infinity generalizing previous results of gupta and hero to the case of nonindependent observations if d represents the dimension of one sample it is proved that the error exponent converges at rate n2 d to the one obtained in the absence of quantization as an application relevant high rate quantization strategies which lead to a large error exponent are determined numerical results indicate that the proposed quantization rule can yield better performance than existing ones in terms of detection error

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

radial basis function network training using a nonsymmetric partition of the input space and particle swarm optimization this paper presents a novel algorithm for training radial basis function rbf networks in order to produce models with increased accuracy and parsimony the proposed methodology is based on a nonsymmetric variant of the fuzzy means fm algorithm which has the ability to determine the number and locations of the hidden node rbf centers whereas the synaptic weights are calculated using linear regression taking advantage of the short computational times required by the fm algorithm we wrap a particle swarm optimization pso based engine around it designed to optimize the fuzzy partition the result is an integrated framework for fully determining all the parameters of an rbf network the proposed approach is evaluated through its application on 12 real world and synthetic benchmark datasets and is also compared with other neural network training techniques the results show that the rbf network models produced by the pso based nonsymmetric fm algorithm outperform the models produced by the other techniques exhibiting higher prediction accuracies in shorter computational times accompanied by simpler network structures

analysis of multiple antenna systems with finite rate feedback using high resolution quantization theory this paper considers the development of a general framework for the analysis of transmit beamforming methods in multiple antenna systems with finite rate feedback inspired by the results of classical high resolution quantization theory the problem of finite rate quantized communication system is formulated as a general fixed rate vector quantization problem with side information available at the encoder or the quantizer but unavailable at the decoder the framework of the quantization problem is sufficiently general to include quantization schemes with general non mean squared distortion functions and constrained source vectors asymptotic distortion analysis of the proposed general quantization problem is provided by extending the vector version of the bennett s integral specifically tight lower and upper bounds of the average asymptotic distortion are proposed sufficient conditions for the achievability of the distortion bounds are also provided and related to corresponding classical fixed rate quantization problems the proposed general methodology provides a powerful analytical tool to study a wide range of finite rate feedback systems to illustrate the utility of the framework we consider the analysis of a finite rate feedback multiple input single output miso beamforming system over independent and identically distributed i i d rayleigh flat fading channels numerical and simulation results are presented that further confirm the accuracy of the analytical results

multidimensional optical transport based on optimized vector quantization inspired signal constellation design an optimized vector quantization inspired signal constellation design ovq scd suitable for multidimensional optical transport is proposed in which signal constellation radii transformation function is optimized and near uniform distribution of points is achieved the proposed ovq scd is used in a tandem with a hybrid multidimensional coded modulation scheme employing slepian sequences as electrical discrete time basis functions orthogonal prolate spheroidal wave functions as impulse responses of optical filters in orthogonal division multiplexing and spatial modes as optical continuous time basis functions it has been shown that the proposed multidimensional coded modulation schemes based on ovq scds outperform corresponding counterparts and can be used to enable beyond 10 pb s serial optical transport over spatial division multiplexing sdm fibers as well as beyond 1 pb s transport over smfs the proposed ovq scd based hybrid multidimensional coded modulation scheme can simultaneously solve the problems related to the limited bandwidth of information infrastructure high energy consumption and heterogeneity of network segments while enabling elastic and dynamic bandwidth allocation

commuting graphs of full matrix rings over finite fields abstract let r be a non commutative ring and z r be its center the commuting graph of r is defined to be the graph r whose vertex set is r z r and two distinct vertices are joint by an edge whenever they commute let f be a finite field n 2 an arbitrary integer and r be a ring with identity such that r m n f where m n f is the ring of n n matrices over f here we prove that r m n f we also show that if f is prime and n 2 then r m 2 f

a hierarchical rbf online learning algorithm for real time 3 d scanner in this paper a novel real time online network model is presented it is derived from the hierarchical radial basis function hrbf model and it grows by automatically adding units at smaller scales where the surface details are located while data points are being collected real time operation is achieved by exploiting the quasi local nature of the gaussian units through the definition of a quad tree structure to support their receptive field local network reconfiguration can be obtained the model has been applied to 3 d scanning where an updated real time display of the manifold to the operator is fundamental to drive the acquisition procedure itself quantitative results are reported which show that the accuracy achieved is comparable to that of two batch approaches batch hrbf and support vector machines svms however these two approaches are not suitable to real time online learning moreover proof of convergence is also given

fast and efficient second order method for training radial basis function networks this paper proposes an improved second order iso algorithm for training radial basis function rbf networks besides the traditional parameters including centers widths and output weights the input weights on the connections between input layer and hidden layer are also adjusted during the training process more accurate results can be obtained by increasing variable dimensions initial centers are chosen from training patterns and other parameters are generated randomly in limited range taking the advantages of fast convergence and powerful search ability of second order algorithms the proposed iso algorithm can normally reach smaller training testing error with much less number of rbf units during the computation process quasi hessian matrix and gradient vector are accumulated as the sum of related sub matrices and vectors respectively only one jacobian row is stored and used for multiplication instead of the entire jacobian matrix storage and multiplication memory reduction benefits the computation speed and allows the training of problems with basically unlimited number of patterns several practical discrete and continuous classification problems are applied to test the properties of the proposed iso training algorithm

learning subspace based rbfnn using coevolutionary algorithm for complex classification tasks many real world classification problems are characterized by samples of a complex distribution in the input space the classification accuracy is determined by intrinsic properties of all samples in subspaces of features this paper proposes a novel algorithm for the construction of radial basis function neural network rbfnn classifier based on subspace learning in this paper feature subspaces are obtained for every hidden node of the rbfnn during the learning process the connection weights between the input layer and the hidden layer are adjusted to produce various subspaces with dominative features for different hidden nodes the network structure and dominative features are encoded in two subpopulations that are cooperatively coevolved using the coevolutionary algorithm to achieve a better global optimality for the estimated rbfnn experimental results illustrate that the proposed algorithm is able to obtain rbfnn models with both better classification accuracy and simpler network structure when compared with other learning algorithms thus the proposed model provides a more flexible and efficient approach to complex classification tasks by employing the local characteristics of samples in subspaces

regularization in the selection of radial basis function centers subset selection and regularization are two well known techniques that can improve the generalization performance of nonparametric linear regression estimators such as radial basis function networks this paper examines regularized forward selection rfs a combination of forward subset selection and zero order regularization an efficient implementation of rfs into which either delete 1 or generalized cross validation can be incorporated and a reestimation formula for the regularization parameter are also discussed simulation studies are presented that demonstrate improved generalization performance due to regularization in the forward selection of radial basis function centers

on the diameters of commuting graphs the commuting graph of a ring r denoted by r is a graph whose vertices are all non central elements of r and two distinct vertices x and y are adjacent if and only if xy yx let d be a division ring and n 3 in this paper we investigate the diameters of mn d and determine the diameters of some induced subgraphs of mn d such as the induced subgraphs on the set of all non scalar non invertible nilpotent idempotent and involution matrices in mn d for every field f it is shown that if mn f is a connected graph then diam mn f 6 we conjecture that if mn f is a connected graph then diam mn f 5 we show that if f is an algebraically closed field or n is a prime number and mn f is a connected graph then diam mn f 4 finally we present some applications to the structure of pairs of idempotents which may prove of independent interest

vanishing distortion and shrinking cells we establish an asymptotic connection between vanishing rth power distortion and shrinking cell diameters for vector quantizers with convex cells as a consequence a number of shrinking cell conditions may be easily verified by showing that the quantizers in question have distortion tending to zero this also plays an important role in the asymptotic analysis of a common greedy growing scheme for tree structured vector quantizers

algebraic feature extraction of image for recognition abstract the extraction of image features is one of the fundamental tasks in image recognition up until now there have been several kinds of features to be used for the purpose of image recognition as follows 1 visual features 2 statistical features of pixel 3 transform coefficient features in addition there is another kind of feature which the author believes is very useful i e 4 algebraic features which represent intrinsic attributions of an image singular values sv of image are this kind of feature in this paper we prove that sv feature vector has some important properties of algebraic and geometric invariance and insensitiveness to noise these properties are very useful for the description and recognition of images as an example sv feature vector is used for the problem of recognizing human facial images in this paper using sv feature vector samples of facial images a normal pattern bayes classification model based on sammon s optimal descriminant plane is constructed the experimental result shows that sv feature vector has good performance of class separation

gradient radial basis function networks for nonlinear and nonstationary time series prediction we present a method of modifying the structure of radial basis function rbf network to work with nonstationary series that exhibit homogeneous nonstationary behavior in the original rbf network the hidden node s function is to sense the trajectory of the time series and to respond when there is a strong correlation between the input pattern and the hidden node s center this type of response however is highly sensitive to changes in the level and trend of the time series to counter these effects the hidden node s function is modified to one which detects and reacts to the gradient of the series we call this new network the gradient rbf grbf model single and multistep predictive performance for the mackey glass chaotic time series were evaluated using the classical rbf and grbf models the simulation results for the series without and with a tine varying mean confirm the superior performance of the grbf predictor over the rbf predictor

nonlinear image upsampling method based on radial basis function interpolation in this paper we present a novel edge directed upsampling method based on radial basis function rbf interpolation in order to remove artifacts such as blurred edges or blocking effects we suggest a nonlinear method capable of taking edge information into account the resampling evaluation is determined according to the edge orientation the proposed scheme is as simple to implement as linear methods but it demonstrates improved visual quality by preserving the edge features better than the classical linear interpolation methods the algorithm is compared with some well known linear schemes as well as recently developed nonlinear schemes the resulting images demonstrate the new algorithm s ability to magnify an image while preserving the edge features

bennett s integral for vector quantizers this paper extends bennett s 1948 integral from scalar to vector quantizers giving a simple formula that expresses the rth power distortion of a many point vector quantizer in terms of the number of points point density function inertial profile and the distribution of the source the inertial profile specifies the normalized moment of inertia of quantization cells as a function of location the extension is formulated in terms of a sequence of quantizers whose point density and inertial profile approach known functions as the number of points increase precise conditions are given for the convergence of distortion suitably normalized to bennett s integral previous extensions did not include the inertial profile and consequently provided only bounds or applied only to quantizers with congruent cells such as lattice and optimal quantizers the new version of bennett s integral provides a framework for the analysis of suboptimal structured vector quantizers it is shown how the loss in performance of such quantizers relative to optimal unstructured ones can be decomposed into point density and cell shape losses as examples these losses are computed for product quantizers and used to gain further understanding of the performance of scalar quantizers applied to stationary memoryless sources and of transform codes applied to gaussian sources with memory it is shown that the short coming of such quantizers is that they must compromise between point density and cell shapes

two results on the asymptotic performance of quantizers a necessary and sufficient condition is presented for the normalized asymptotic r th power distortion of a mismatched multidimensional quantizer to converge to a certain optimum constant known as bennett s integral the dimensionality of our results is allowed to approach infinity in order to make some universal source coding comparisons to quantization theory

cooperative competitive genetic evolution of radial basis function centers and widths for time series prediction in a radial basis function rbf network the rbf centers and widths can be evolved by a cooperative competitive genetic algorithm the set of genetic strings in one generation of the algorithm represents one rep network not a population of competing networks this leads to moderate computation times for the algorithm as a whole selection operates on individual rbfs rather than on whole networks selection therefore requires a genetic fitness function that promotes competition among rbfs which are doing nearly the same job while at the same time promoting cooperation among rbfs which cover different parts of the domain of the function to be approximated niche creation resulting from a fitness function of the form w sub i sup spl beta e w sub i sup spl beta 1 spl beta 2 can facilitate the desired cooperative competitive behavior the feasibility of the resulting algorithm to evolve networks of gaussian inverse multiquadric and thin plate spline rbfs is demonstrated by predicting the mackey glass time series for each type of rbf and for networks of 25 50 75 100 125 and 150 rbf units prediction errors for the evolved gaussian rbf networks are 50 70 lower than rbf networks obtained by k means clustering

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

a simple class of asymptotically optimal quantizers a simple class of quantizers is introduced which are asymptotically optimal as the number of quantization levels increases to infinity with respect to a mean r th power absolute error distortion measure these asymptotically optimal quantizers are very easy to compute their performance is evaluated for several distributions and compares favorably with the performance of the optimal quantizers in all cases for which the latter have been computed in addition their asymptotic robustness is studied under location scale and shape mismatch for several families of distributions

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

on diameter of components in commuting graphs abstract for each prime p 7 we construct two matrices inside m 2 p q such that their distance in a commuting graph m 2 p q equals six

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

sparse multivariate gaussian mixture regression fitting a multivariate gaussian mixture to data represents an attractive as well as challenging problem in especial when sparsity in the solution is demanded achieving this objective requires the concurrent update of all parameters weight centers and precisions of all multivariate gaussian functions during the learning process such is the focus of this paper which presents a novel method founded on the minimization of the error of the generalized logarithmic utility function gluf this choice which allows us to move smoothly from the mean square error mse criterion to the one based on the logarithmic error yields an optimization problem that resembles a locally convex problem and can be solved with a quasi newton method the gluf framework also facilitates the comparative study between both extremes concluding that the classical mse optimization is not the most adequate for the task the performance of the proposed novel technique is demonstrated on simulated as well as realistic scenarios

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