yang mills equations on the two dimensional sphere we construct a 1 1 correspondence between the equivalence classes of yang mills fields overs2 and the conjugacy classes of closed geodesics of the structure group furthermore we give an explicit isolation theorem for any yang mills field overs2

convergence of vector quantizers with applications to optimal quantization suppose that a sequence of probability distribution functions f n converges weakly to a distribution function f does the sequence of optimal quantizers for the f n s converge to an optimal quantizer for f if so do the respective distortions converge to the optimal distortion for f sufficient conditions are given to guarantee the convergence for both scalar and vector quantizers with a general class of distortion measures these results are used to prove the existence of minimum rth power distortion vector quantizers and the convergence of a proposed algorithm for constructing optimal quantizers

optimal grouping for group minimax hypothesis testing bayesian hypothesis testing and minimax hypothesis testing represent extreme instances of detection in which the prior probabilities of the hypotheses are either completely and precisely known or are completely unknown group minimax also known as gamma minimax is a robust intermediary between bayesian and minimax hypothesis testing that allows for coarse or partial advance knowledge of the hypothesis priors by using information on sets in which the prior lies existing work on group minimax however does not consider the question of how to define the sets or groups of priors it is assumed that the groups are given in this paper we propose a novel intermediate detection scheme formulated through the quantization of the space of prior probabilities that optimally determines groups and also representative priors within the groups we show that when viewed from a quantization perspective group minimax amounts to determining centroids with a minimax bayes risk error divergence distortion criterion the appropriate bregman divergence for this task in addition the optimal partitioning of the space of prior probabilities is a bregman voronoi diagram together the optimal grouping and representation points are an epsilon net with respect to bayes risk error divergence and permit a rate distortion type asymptotic analysis of detection performance with the number of groups examples of detecting signals corrupted by additive white gaussian noise and of distinguishing exponentially distributed signals are presented

asymptotically optimal quantizers for detection of i i d the asymptotic probability of error for quantization in maximum likelihood tests is analyzed the authors assume quantizers with large numbers of levels generated from a companding function a theorem that relates the companding function to the asymptotic probability of error is proved the companding function is then optimized

monotony of lloyd s method ii for log concave density and convex error weighting function corresp kieffer recently showed that for a log concave probability density there is a unique n level quantizer which is a stationary point of the expected value of an increasing convex and continuously differentiable error weighting function dependent on the absolute quantization error it is shown that the requirement of continuous differentiability can be dropped using the monotony of lloyd s method ii the proof of which is based on a generalization of previous work by the author

a survey of the theory of source coding with a fidelity criterion the purpose of this paper is threefold 1 to acquaint the reader with the types of problems that have been considered in the area of source coding with a fidelity criterion 2 to survey results that have been obtained on these problems and 3 to outline future research trends in the area

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

uniqueness of locally optimal quantizer for log concave density and convex error weighting function it is desired to encode a random variable x using an n level quantizer q to minimize the expected distortion e rho x q x i where the error weighting function rho is convex strictly increasing and continuously differentiable it is shown that if x has a log concave density then there exists a unique locally optimal quantizer q ast and lloyd s method i may be used to find q ast trushkin had earlier shown this result for the error weighting functions rho t equiv t and rho t euiv t 2

blow up of solutions to parabolic equations with nonstandard growth conditions we study the phenomenon of finite time blow up in solutions of the homogeneous dirichlet problem for the parabolic equation u t div a x t u p x email protected u b x t u s x t 2u with variable exponents of nonlinearity p x s x t 1 two different cases are studied in the case of semilinear equation with p x 2 a x t 1 b x t b 0 we show that the finite time blow up happens if the initial function is sufficiently large and either min email protected x t email protected t 2 for all t 0 or s t 2 s t 2 as t and 1 e s 2 s s ds b 0 a t x t 0 b t x t 0 email protected x 2 and maxp x email protected x

quantization via empirical divergence maximization empirical divergence maximization edm refers to a recently proposed strategy for estimating f divergences and likelihood ratio functions this paper extends the idea to empirical vector quantization where one seeks to empirically derive quantization rules that maximize the kullback leibler divergence between two statistical hypotheses we analyze the estimator s error convergence rate leveraging tsybakov s margin condition and show that rates as fast as n 1 are possible where n equals the number of training samples we also show that the flynn and gray algorithm can be used to efficiently compute edm estimates and show that they can be efficiently and accurately represented by recursive dyadic partitions the edm formulation have several advantages first the formulation gives access to the tools and results of empirical process theory that quantify the estimator s error convergence rate second the formulation provides a previously unknown derivation for the flynn and gray algorithm third the flexibility it affords allows one to avoid a small cell assumption common in other approaches finally we illustrate the potential use of the method through an example

blow up for parabolic and hyperbolic problems with variable exponents abstract in this paper we study the blow up problem for positive solutions of parabolic and hyperbolic problems with reaction terms of local and nonlocal type involving a variable exponent we prove the existence of initial data such that the corresponding solutions blow up at a finite time

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

existence of solutions for semilinear neutral stochastic functional differential equations with nonlocal conditions abstract in this paper by using fractional power of operators and sadovskii fixed point theorem the existence of mild and strong solutions of semilinear neutral functional differential evolution equations with nonlocal conditions is studied the results are a generalization and continuation of the recent results on this issue an example is provided to illustrate the theory

some inequalities for information divergence and related measures of discrimination inequalities which connect information divergence with other measures of discrimination or distance between probability distributions are used in information theory and its applications to mathematical statistics ergodic theory and other scientific fields we suggest new inequalities of this type often based on underlying identities as a consequence we obtain certain improvements of the well known pinsker inequality our study depends on two measures of discrimination called capacitory discrimination and triangular discrimination the discussion contains references to related research and comparison with other measures of discrimination e g ali silvey csiszar 1996 1966 divergences and in particular the hellinger distance

loop groups and yang mills theory in dimension two given a connection in ag bundle overs2 then a process called radial trivialization from the poles gives a unique clutching function i e an element of the loop group g up to gauge equivalence is completely determined by and a map f s2 g into the lie algebra moreover the yang mills function of is the sum of the energy of and the square of a certain norm off in particular the yang mills functional has the same morse theory as the energy functional on g there is a similar description of connections in ag bundle over an arbitrary riemann surface but so far not of the yang mills functional

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

the non regular ceo problem we consider the ceo problem for non regular source distributions such as uniform or truncated gaussian a group of agents observe independently corrupted versions of data and transmit coded versions over rate limited links to a ceo the ceo then estimates the underlying data based on the received coded observations agents are not allowed to convene before transmitting their observations this formulation is motivated by the practical problem of a firm s ceo estimating non regular beliefs about a sequence of events before acting on them agents observations are modeled as jointly distributed with the underlying data through a given conditional probability density function we study the asymptotic behavior of the minimum achievable mean squared error distortion at the ceo in the limit when the number of agents l and the sum rate r tend to infinity we establish a 1 r 2 convergence of the distortion an intermediate regime of performance between the exponential behavior in discrete ceo problems berger zhang and viswanathan 1996 and the 1 r behavior in gaussian ceo problems viswanathan and berger 1997 achievability is proved by a layered architecture with scalar quantization distributed entropy coding and midrange estimation the converse is proved using the bayesian chazan zakai ziv bound

vector quantizer design for memoryless gaussian gamma and laplacian sources locally optimum vector quantizer vq designs are presented for memoryless gaussian gamma and laplacian sources for gaussian sources low 2 6 dimensional vector quantization provides relatively little improvement in mean squared error mse compared to the minimum mean squared error mmse scalar quantizer for laplacian or gamma sources however significant improvement in mse is available with vector quantization the laplacian and gamma 6 bit sixdimensional vector quantizers achieve respectively improvements of 2 and 4 5 db over the corresponding scalar mmse quantizer distortions

k vectors an alternating minimization algorithm for learning regression functions the k vectors algorithm for learning regression functions proposed here is akin to the well known k means algorithm both algorithms partition the feature space but unlike the k means algorithm the k vectors algorithm aims to reconstruct the response rather than the feature the partitioning rule of the algorithm is based on maximizing the correlation inner product of the feature vector with a set of k vectors and generates polyhedral cells similar to the ones generated by the nearest neighbor rule of the k means algorithm similarly to k means the learning algorithm alternates between two types of steps in the first type of steps k labels are determined via a centroid type rule in the response space which uses a surrogate hinge type loss function to the mean squared error loss function in the second type of steps the k vectors which determine the partition are updated according to a multiclass classification rule in the spirit of support vector machines it is proved that both steps of the algorithm only require solving convex optimization problems and that the algorithm is empirically consistent as the length of the training sequence increases to infinity fixed points of the empirical version of the algorithm tend to fixed points of the population version of the algorithm learnability of the predictor class posit by the algorithm is also established

sufficient conditions for uniqueness of a locally optimal quantizer for a class of convex error weighting functions sufficient conditions are presented for uniqueness of a locally optimal quantizer being a stationary point of the quantization distortion measure e f x eta the expected value of an error weighting function f x eta where x is a random variable to be quantized where the probability density function p x describing x is continuous and positive on some finite or infinite interval and zero outside it and where eta is the quantization error the function f x eta is assumed convex and symmetric in eta and zero only for eta 0 it is shown that in the cases of f x eta eta 2 and f x eta eta the simple condition of concavity of in p x is sufficient for uniqueness of a locally optimal quantizer

the yang mills measure for s2 abstract the yang mills measure for gauge fields over the two sphere is constructed using a conditioned white noise process stochastic differential equations are used to study parallel translation under this measure expectation values of a broad class of wilson loop configurations are computed explicitly and it is shown that these values are invariant under area preserving diffeomorphisms of the sphere

termination and continuity of greedy growing for tree structured vector quantizers tree structured vector quantizers tsvq provide a computationally efficient variable rate method of compressing vector valued data in applications the problem of designing a tsvq from empirical training data is critical greedy growing algorithms are a common and effective approach to the design problem they are recursive procedures that produce a tsvq one node at a time by optimizing a simple splitting criterion at each step while unsupervised greedy growing algorithms are well understood from an experimental point of view there has been little theory to support their use or to examine their behavior on large training sets the authors present a rigorous analysis of a greedy growing algorithm proposed by riskin 1990 riskin and gray 1991 and balakrishnan 1991 the first part of the paper is a description of the algorithm and an examination of its asymptotic behavior as it applies to a fixed absolutely continuous distribution the second part of the paper establishes the structural consistency of the algorithm with respect to a convergent sequence of distributions as an application the authors obtain results concerning the large sample empirical behavior of the algorithm when it is applied to stationary ergodic training vectors

the complexity of the generalized lloyd max problem corresp a simple combinatorial special case of the generalized lloyd max or quantization problem is shown to be nondeterministic polynomial np complete em a fortiori the general problem of communication theory in its combinatorial forms has at least that complexity

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

the semiclassical limit for gauge theory on s 2 it is shown that the yang mills measurez h 1 e s h d whereh 0 describing gauge fields on the two sphere converges to a probability measure on the moduli space of yang mills connections ons2 ash 0

global convergence and empirical consistency of the generalized lloyd algorithm the generalized lloyd algorithm for vector quantizer design is analyzed as a descent algorithm for nonlinear programming a broad class of convex distortion functions is considered and any input distribution that has no singular continuous part is allowed a well known convergence theorem is applied to show that iterative applications of the algorithm produce a sequence of quantizers that approaches the set of fixed point quantizers the methods of the theorem are extended to sequences of algorithms yielding results on the behavior of the algorithm when an unknown distribution is approximated by a training sequence of observations it is shown that as the length of the training sequence grows large that 1 fixed point quantizers for the training sequence approach the set of fixed point quantizers for the true distribution and 2 limiting quantizers produced by the algorithm with the training sequence distribution perform no worse than limiting quantizers produced by the algorithm with the true distribution

encoding of correlated observations an important class of engineering problems involves sensing an environment and making estimates based on the phenomena sensed in the traditional model of this problem the sensors observations are available to the estimator without alteration there is growing interest in em distributed sensing systems in which several observations are communicated to the estimator over channels of limited capacity the observations must be separately encoded so that the target can be estimated with minimum distortion two questions are addressed for a special case of this problem wherein there are two sensors which observe noisy data and communicate with a single estimator 1 if the encoder is unlimited in complexity what communication rates and distortions can be achieved 2 if the encoder must be a quantizer a mapping of a single observation sample into a digital output how can it be designed for good performance the first question is treated by the techniques of information theory it is proved that a given combination of rates and distortion is achievable if there exist degraded versions of the observations that satisfy certain formulas the second question is treated by two approaches in the first the outputs of the quantizers undergo a second stage of encoding which exploits their correlation to reduce the output rate algorithms which design the second stage are presented and tested the second approach is based on the em distributional distance a measure of dissimilarity between two probability distributions an algorithm to modify a quantizer for increased distributional distance is derived and tested

combining image compression and classification using vector quantization we describe a method of combining classification and compression into a single vector quantizer by incorporating a bayes risk term into the distortion measure used in the quantizer design algorithm once trained the quantizer can operate to minimize the bayes risk weighted distortion measure if there is a model providing the required posterior probabilities or it can operate in a suboptimal fashion by minimizing the squared error only comparisons are made with other vector quantizer based classifiers including the independent design of quantization and minimum bayes risk classification and kohonen s lvq a variety of examples demonstrate that the proposed method can provide classification ability close to or superior to learning vq while simultaneously providing superior compression performance

information theoretic bounds on target recognition performance based on degraded image data this paper derives bounds on the performance of statistical object recognition systems wherein an image of a target is observed by a remote sensor detection and recognition problems are modeled as composite hypothesis testing problems involving nuisance parameters we develop information theoretic performance bounds on target recognition based on statistical models for sensors and data and examine conditions under which these bounds are tight in particular we examine the validity of asymptotic approximations to probability of error in such imaging problems problems involving gaussian poisson and multiplicative noise and random pixel deletions are considered as well as least favorable gaussian clutter a sixth application involving compressed sensor image data is considered in some detail this study provides a systematic and computationally attractive framework for analytically characterizing target recognition performance under complicated non gaussian models and optimizing system parameters

quantization of prior probabilities for collaborative distributed hypothesis testing this paper studies the quantization of prior probabilities drawn from an ensemble in distributed detection with data fusion by combination of binary local decisions design and performance equivalences between a team of n agents and a more powerful single agent are obtained effects of identical quantization and diverse quantization on mean bayes risk are compared it is shown that when agents using diverse quantizers interact to agree on a perceived common risk the effective number quantization levels is increased with this collaboration optimal diverse regular quantization with k cells per quantizer performs as well as optimal identical quantization with n k 1 1 cells per quantizer similar results are obtained for the maximum bayes risk error criterion

nearest neighbor clustering a baseline method for consistent clustering with arbitrary objective functions clustering is often formulated as a discrete optimization problem the objective is to find among all partitions of the data set the best one according to some quality measure however in the statistical setting where we assume that the finite data set has been sampled from some underlying space the goal is not to find the best partition of the given sample but to approximate the true partition of the underlying space we argue that the discrete optimization approach usually does not achieve this goal and instead can lead to inconsistency we construct examples which provably have this behavior as in the case of supervised learning the cure is to restrict the size of the function classes under consideration for appropriate small function classes we can prove very general consistency theorems for clustering optimization schemes as one particular algorithm for clustering with a restricted function space we introduce nearest neighbor clustering similar to the k nearest neighbor classifier in supervised learning this algorithm can be seen as a general baseline algorithm to minimize arbitrary clustering objective functions we prove that it is statistically consistent for all commonly used clustering objective functions

quantization of prior probabilities for hypothesis testing in this paper bayesian hypothesis testing is investigated when the prior probabilities of the hypotheses taken as a random vector are quantized nearest neighbor and centroid conditions are derived using mean bayes risk error mbre as a distortion measure for quantization a high resolution approximation to the distortion rate function is also obtained human decision making in segregated populations is studied assuming bayesian hypothesis testing with quantized priors

on source coding with side information dependent distortion measures high resolution bounds in lossy coding of a real memoryless source are considered when side information is present let x be a smooth source and let y be the side information first we treat the case when both the encoder and the decoder have access to y and we establish an asymptotically tight high resolution formula for the conditional rate distortion function r sub x y d for a class of locally quadratic distortion measures which may be functions of the side information we then consider the case when only the decoder has access to the side information i e the wyner ziv problem for side information dependent distortion measures we give an explicit formula which tightly approximates the wyner ziv rate distortion function r sup wz d for small d under some assumptions on the joint distribution of x and y these results demonstrate that for side information dependent distortion measures the rate loss r sup wz d r sub x y d can be bounded away from zero in the limit of small d this contrasts the case of distortion measures which do not depend on the side information where the rate loss vanishes as d spl rarr 0

hierarchical dirichlet processes we consider problems involving groups of data where each observation within a group is a draw from a mixture model and where it is desirable to share mixture components between groups we assume that the number of mixture components is unknown a priori and is to be inferred from the data in this setting it is natural to consider sets of dirichlet processes one for each group where the well known clustering property of the dirichlet process provides a nonparametric prior for the number of mixture components within each group given our desire to tie the mixture models in the various groups we consider a hierarchical model specifically one in which the base measure for the child dirichlet processes is itself distributed according to a dirichlet process such a base measure being discrete the child dirichlet processes necessarily share atoms thus as desired the mixture models in the different groups necessarily share mixture components we discuss representations of hierarchical dirichlet processes

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

fine quantization in signal detection and estimation the performance lost to data quantization in signal detection and estimation procedures is considered the performance is measured by f divergences which are useful indices of discrimination information between statistical hypotheses some properties of the f divergences are briefly discussed and a general result for the loss in divergence due to uniform data quantization is obtained several applications of this result in specific problems of signal detection and estimation being developed and some numerical results that illustrate the asymptotic behavior of the divergence in these applications are given the divergence lost to finite nonuniform quantization through the use of a companding model is considered and the asymptotically optimum nonuniform quantizer within this class is derived some interesting problems that remain open are discussed

collaborative human decision making with random local thresholds this paper considers a collaborative human decision making framework in which local decisions made at the individual agents are combined at a moderator to make the final decision more specifically we consider a binary hypothesis testing problem in which a group of n people makes individual decisions on which hypothesis is true based on a threshold based scheme and the thresholds are modeled as random variables we assume that in general the decisions are not received by the moderator perfectly and the communication errors are modeled via a binary asymmetric channel assuming that the moderator does not have the knowledge of exact values of thresholds used by the individual decision makers but has probabilistic information the performance in terms of the probability of error of the likelihood ratio based decision fusion scheme is derived when there are two agents in the decision making system we show that the statistical parameters of the threshold distributions have optimal set of values which result in the minimum probability of error and we analytically derive these optimal values under certain conditions we further provide detailed performance comparison to the case where the likelihood ratio based decision fusion is performed at the moderator with exact knowledge of the thresholds used by individual agents for an arbitrary number of human agents n 2 we derive the performance of decision fusion with majority rule using certain approximations when the individual thresholds are modeled as random variables

optimal binary vector quantization via enumeration of covering codes binary vector quantization bvq refers to block coding of binary vectors under a fidelity measure covering codes were studied as a means of lattice bvq but a further source coding problem hidden in the equivalence of covering codes has seemingly eluded attention given a d dimensional hypercube code space equivalent covering codes of the same covering radius but of different codewords have different expected bvq distortions for a general probability mass function thus one can minimize within the code equivalence the expected distortion over all different covering codes this leads a two stage optimization scheme for bvq design first we use an optimal covering code to minimize the maximum per vector distortion at a given rate then under the minmax constraint we minimize the expected quantization distortion this minmax constrained bvq method mcbvq controls both the maximum and average distortions and hence improves subjective quality of compressed binary images mcbvq also avoids poor local minima that may trap the generalized lloyd method the 7 4 hamming code and 8 4 extended hamming code are found to be particularly suitable for mcbvq on binary images an efficient and simple algorithm is introduced to enumerate all distinct 7 4 hamming 8 4 extended hamming codes and compute the corresponding expected distortions in optimal mcbvq design furthermore mcbvq using linear covering codes has a compact codebook and a fast syndrome encoding algorithm

complexity constrained trellis quantizers a reasonable measure of quantizer complexity is the expected number of quanta per input sample for which distortion is computed given this measure a rate distortion complexity theory is obtained by extending earlier work in alphabet constrained rate distortion theory numerical results show that operation on the alphabet constrained rate distortion bound can be obtained with a complexity of two furthermore lloyd max conditions are shown to describe the minimum of a slightly constrained version of the rate distortion complexity problem complexity constrained design methods are applied first to trellis coded quantizers where they are shown to reduce arithmetic operations by at least 25 they are then used to develop model based trellis quantizers the trellises of which are derived from a markov model of the source simulation results confirm that excellent performance can be obtained with modest complexity

linear precoders for the detection of a gaussian process in wireless sensors networks we investigate the performance of neyman pearson detection of a stationary gaussian process in noise using a large wireless sensor network wsn in our model each sensor compresses its observation sequence using a linear precoder and a final decision is taken by a fusion center fc based on the compressed information two families of precoders are studied random i i d precoders and orthogonal precoders we analyse their performance under a regime where both the number of sensors k and the number of samples n per sensor tend to infinity at the same rate that is k n c 0 1 contributions are as follows 1 using results from random matrix theory and large toeplitz matrices we prove that when the above families of precoders are used the miss probability of the neyman pearson detector converges exponentially to zero closed form expressions of the corresponding error exponents are derived 2 in particular we propose a practical orthogonal precoding strategy the principal frequencies strategy pfs which achieves the best error exponent among all orthogonal strategies and which requires very little signaling overhead between the central processor and the nodes of the network 3 when the pfs is used a simplified low complexity testing procedure can be implemented at the fc we show that the proposed suboptimal test enjoys the same error exponent as the neyman pearson test which indicates a similar asymptotic behavior of the performance we illustrate our findings by numerical experiments on several examples

optimum quantization for detection vector quantization is a procedure widely used in communications the problem has been solved for the basic detection situation using the locally optimal procedure and with some constraints on the quantization scheme it is stated in its widest sense which consists of finding the partition of the observation space and the values associated with it in such a way that the detection performance is optimal it is solved completely with the deflection criterion and the solution introduces the concept of quantization by a likelihood ratio procedure

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

quantization and the method of k means asymptotic results from the statistical theory of k means clustering are applied to problems of vector quantization the behavior of quantizers constructed from long training sequences of data is analyzed by relating it to the consistency problem for k means

bayes risk weighted vector quantization with posterior estimation for image compression and classification classification and compression play important roles in communicating digital information their combination is useful in many applications including the detection of abnormalities in compressed medical images in view of the similarities of compression and low level classification it is not surprising that there are many similar methods for their design because some of these methods are useful for designing vector quantizers it seems natural that vector quantization vq is explored for the combined goal we investigate several vq based algorithms that seek to minimize both the distortion of compressed images and errors in classifying their pixel blocks these algorithms are investigated with both full search and tree structured codes we emphasize a nonparametric technique that minimizes both error measures simultaneously by incorporating a bayes risk component into the distortion measure used for the design and encoding we introduce a tree structured posterior estimator to produce the class posterior probabilities required for the bayes risk computation in this design for two different image sources we demonstrate that this system provides superior classification while maintaining compression close or superior to that of several other vq based designs including kohonen s 1992 learning vector quantizer and a sequential quantizer classifier design

adaptive learning vector quantization for online parametric estimation this paper addresses the problem of parameter estimation in a quantized and online setting a sensing unit collects random vector valued samples from the environment these samples are quantized and transmitted to a central processor which generates an online estimate of the unknown parameter this paper provides a closed form expression of the excess mean square error mse caused by quantization in the high rate regime i e when the number of quantization levels is supposed to be large next we determine the quantizers which mitigate the excess mse the optimal quantization rule unfortunately depends on the unknown parameter to circumvent this issue we introduce a novel adaptive learning vector quantization scheme which allows to simultaneously estimate the parameter of interest and select an efficient quantizer

type based estimation over multiaccess channels we study the problem of communicating sensor readings over a gaussian multiaccess channel we focus on the scenario that each sensor observes a single random variable and transmits it using certain signaling in a shared channel the objective is the design of channel waveforms i e the signal constellation to facilitate the estimation of field parameters from the channel output we propose a communication scheme in which sensors transmit according to the type of their observations type based multiple access tbma and show that the tbma is asymptotically optimal in the limit of large number of sensors if the sensor channel gains are identical in particular we show that tbma together with a variant of the maximum likelihood estimator achieves the cramer rao bound asymptotically we then extend the asymptotic analysis of tbma to fading channels and compare the performance of tbma with other orthogonal allocation methods such as time division multiple access

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

high resolution source coding for non difference distortion measures multidimensional companding entropy coded vector quantization is studied using high resolution multidimensional companding over a class of nondifference distortion measures for distortion measures which are locally quadratic a rigorous derivation of the asymptotic distortion and entropy coded rate of multidimensional companders is given along with conditions for the optimal choice of the compressor function this optimum compressor when it exists depends on the distortion measure but not on the source distribution the rate distortion performance of the companding scheme is studied using an asymptotic expression for the rate distortion function which parallels the shannon lower bound for difference distortion measures it is proved that the high resolution performance of the scheme is arbitrarily close to the rate distortion limit for large quantizer dimensions if the compressor function and the lattice quantizer used in the companding scheme are optimal extending an analogous statement for entropy coded lattice quantization and mse distortion the companding approach is applied to obtain a high resolution quantizing scheme for noisy sources

second critical exponent and life span for pseudo parabolic equation this paper studies the second critical exponent and life span of solutions for the pseudo parabolic equation ut k ut u up in rn 0 t with p 1 k 0 it is proved that the second critical exponent i e the decay order of the initial data required by global solutions in the coexistence region of global and non global solutions is independent of the pseudo parabolic parameter k nevertheless it is revealed that the viscous term k ut relaxes restrictions on the amplitude of the initial data required by the global solutions moreover it is observed that the life span of the non global solutions will be delayed by the third order viscous term finally some numerical examples are given to illustrate all these results

convergence of the lloyd algorithm for computing centroidal voronoi tessellations centroidal voronoi tessellations cvts are voronoi tessellations of a bounded geometric domain such that the generating points of the tessellations are also the centroids mass centers of the corresponding voronoi regions with respect to a given density function centroidal voronoi tessellations may also be defined in more abstract and more general settings due to the natural optimization properties enjoyed by cvts they have many applications in diverse fields the lloyd algorithm is one of the most popular iterative schemes for computing the cvts but its theoretical analysis is far from complete in this paper some new analytical results on the local and global convergence of the lloyd algorithm are presented these results are derived through careful utilization of the optimization properties shared by cvts numerical experiments are also provided to substantiate the theoretical analysis

on the critical exponent for reaction diffusion equations in this paper we study the initial boundary value problem for u t u h t u p with homogeneous dirichlet boundary conditions where h t t q for large t let s sup s positive solutions w of u t u such that mathop lim sup limits t to infty t s parallel w cdot t parallel infty infty then for p 1 q 1 s we show if p p there are global positive solutions that decay to zero uniformly for t if 1 p p then all nontrivial solutions blow up in finite time we determine p for some conical domains in r 2 and r 3 a similar result is derived for a bounded domain if h t e t for large t

on the critical exponent for some semilinear reaction diffusion systems on general domains abstract in this study we consider the parabolic systems u t u f t v p v t v f t u q and u t u f t u r v p v t v f t u q v s in 0 t with homogeneous dirichlet boundary condition p q r s 1 and f c 0 the initial data are considered in the space u 0 v 0 c 0 2 u 0 v 0 0 where is a general domain bounded or unbounded with a smooth boundary we find the conditions that guarantee the global existence or the blow up in finite time of nonnegative solutions these conditions are given in terms of the asymptotic behavior of the solution of the homogeneous linear problem u t u 0 in 0

distributed functional scalar quantization simplified distributed functional scalar quantization dfsq theory provides optimality conditions and predicts performance of data acquisition systems in which a computation on acquired data is desired we address two limitations of previous works prohibitively expensive decoder design and a restriction to source distributions with bounded support we show that a much simpler decoder has equivalent asymptotic performance to the conditional expectation estimator studied previously thus reducing decoder design complexity the simpler decoder features decoupled communication and computation blocks moreover we extend the dfsq framework with the simpler decoder to source distributions with unbounded support finally through simulation results we demonstrate that performance at moderate coding rates is well predicted by the asymptotic analysis and we give new insight on the rate of convergence

global and nonglobal existence for a strongly coupled parabolic system on a general domain abstract we consider the parabolic system u t u f t u r v s v t v g t u q v s in 0 t where r n is either an unbounded or bounded domain and f g c 0 we find conditions for the global existence or nonglobal existence which are expressed in terms of the behavior of s t u 0 as t where u t s t u 0 is the solution of the linear problem u t u 0 u 0 u 0

bayes risk error is a bregman divergence in previous work reported in these transactions we proposed a new distortion measure for the quantization of prior probabilities that are used in the threshold of likelihood ratio test detection bayes risk error in this correspondence we show that the bayes risk error is a member of the class of bregman divergences and discuss the implications of this fact

extremal properties of likelihood ratio quantizers m hypotheses and a random variable y with a different probability distribution under each hypothesis are considered a quantizer is applied to form a quantized random variable gamma y the extreme points of the set of possible probability distributions of gamma y as gamma ranges over all quantizers is characterized optimality properties of likelihood ratio quantizers are established for a very broad class of quantization problems including problems involving the maximization of an ali silvey 1966 distance measure and the neyman pearson variant of the decentralized detection problem

nondegeneracy and weak global convergence of the lloyd algorithm in mathbb r d the lloyd algorithm originated in the context of optimal quantization and represents a fixed point iteration for computing an optimal quantizer reducing average distortion at every step it constructs a voronoi partition of the domain and replaces each generator with the centroid of the corresponding voronoi cell optimal quantization is obtained in the case of a centroidal voronoi tessellation cvt which is a special voronoi tessellation of a domain omega in mathbb r d having the property that the generators of the voronoi diagram are also the centers of mass with respect to a given density function rho geq 0 of the corresponding voronoi cells the lloyd iteration is currently the most popular and elegant algorithm for computing cvts and optimal quantizers but many questions remain about its convergence especially in d dimensional spaces d 1 in this paper we prove that any limit point of the lloyd iteration in any dimensional spaces is nondegenerate provided that omega is a convex and bounded set and rho belongs to l 1 omega and is positive almost everywhere this ensures that the fixed point map remains closed and hence the standard theory of descent methods guarantees weak global convergence of the lloyd iteration to the set of nondegenerate fixed point quantizers while previously only conjectured the convergence properties of the lloyd iteration are rigorously justified under such minimal regularity assumptions on the density functional the results presented in this paper go beyond existing convergence theories for cvts and optimal quantization related algorithms and should be of interest to both the mathematical and engineering communities

quantization for decentralized hypothesis testing under communication constraints in a decentralized hypothesis testing network several peripheral nodes observe an environment and communicate their observations to a central node for the final decision the presence of capacity constraints introduces theoretical and practical problems the following problem is addressed given that the peripheral encoders that satisfy these constraints are scalar quantizers how should they be designed in order that the central test to be performed on their output indices is most powerful the scheme is called cooperative design separate encoding since the quantizers process separate observations but have a common goal they seek to maximize a system wide performance measure the bhattacharyya distance of the joint index space as such a criterion is suggested and a design algorithm to optimize arbitrarily many quantizers cyclically is proposed a simplified version of the algorithm namely an independent design separate encoding scheme where the correlation is either absent or neglected for the sake of simplicity is outlined performances are compared through worked examples

distributed scalar quantization for computing high resolution analysis and extensions communication of quantized information is frequently followed by a computation we consider situations of distributed functional scalar quantization distributed scalar quantization of possibly correlated sources followed by centralized computation of a function under smoothness conditions on the sources and function companding scalar quantizer designs are developed to minimize mean squared error mse of the computed function as the quantizer resolution is allowed to grow striking improvements over quantizers designed without consideration of the function are possible and are larger in the entropy constrained setting than in the fixed rate setting as extensions to the basic analysis we characterize a large class of functions for which regular quantization suffices consider certain functions for which asymptotic optimality is achieved without arbitrarily fine quantization and allow limited collaboration between source encoders in the entropy constrained setting a single bit per sample communicated between encoders can have an arbitrarily large effect on functional distortion in contrast such communication has very little effect in the fixed rate setting

application of honey bee mating optimization algorithm on clustering cluster analysis is one of attractive data mining technique that use in many fields one popular class of data clustering algorithms is the center based clustering algorithm k means used as a popular clustering method due to its simplicity and high speed in clustering large datasets however k means has two shortcomings dependency on the initial state and convergence to local optima and global solutions of large problems cannot found with reasonable amount of computation effort in order to overcome local optima problem lots of studies done in clustering over the last decade modeling the behavior of social insects such as ants and bees for the purpose of search and problem solving has been the context of the emerging area of swarm intelligence honey bees are among the most closely studied social insects honey bee mating may also be considered as a typical swarm based approach to optimization in which the search algorithm is inspired by the process of marriage in real honey bee honey bee has been used to model agent based systems in this paper we proposed application of honeybee mating optimization in clustering hbmk means we compared hbmk means with other heuristics algorithm in clustering such as ga sa ts and aco by implementing them on several well known datasets our finding shows that the proposed algorithm works than the best one

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

trellis source code design as an optimization problem the design of time invariant trellis codes for stationary ergodic discrete time sources is cast as an unconstrained nonlinear optimization problem where the objective function and its derivatives are evaluated by simulation using classical real analysis and the ergodic theorem convergence of the sample encoding distortion and its partial derivatives with respect to the code quantization levels to their ensemble average values is investigated it is found that in the common code design situation the expected per symbol distortion and its first derivatives are available and piecewise continuous but second derivative information is unreliable this indicates that efficient optimization should be performed using a nonderivative or first derivative method that does not compute approximate second derivatives to determine search directions

recursive partitioning to reduce distortion adaptive partitioning of a multidimensional feature space plays a fundamental role in the design of data compression schemes most partition based design methods operate in an iterative fashion seeking to reduce distortion at each stage of their operation by implementing a linear split of a selected cell the operation and eventual outcome of such methods is easily described in terms of binary tree structured vector quantizers this paper considers a class of simple growing procedures for tree structured vector quantizers of primary interest is the asymptotic distortion of quantizers produced by the unsupervised implementation of the procedures it is shown that application of the procedures to a convergent sequence of distributions with a suitable limit yields quantizers whose distortion tends to zero analogous results are established for tree structured vector quantizers produced from stationary ergodic training data the analysis is applicable to procedures employing both axis parallel and oblique splitting and a variety of distortion measures the results of the paper apply directly to unsupervised procedures that may be efficiently implemented on a digital computer

a semilinear parabolic problem with variable reaction on a general domain abstract we are concerned with the parabolic equation u t u f t u p x in 0 t with homogeneous dirichlet boundary condition p c f c 0 and is either a bounded or an unbounded domain the initial data is considered in the space u 0 c 0 u 0 0 we find conditions that guarantee the global existence and the blow up in finite time of nonnegative solutions these conditions are given in terms of the asymptotic behavior of the solution of the homogeneous linear problem u t u 0

on the existence of optimal quantizers corresp the existence of optimal n level quantizers is demonstrated for several classes of quantizers including block quantizers memoryless quantizers and uniform step size quantizers no assumptions whatsoever are made regarding the input probability distribution the distortion measure considered generalizes many of the commonly used error based distortion measures including mean squared error distortion and r th law distortion

k harmonic means data clustering with simulated annealing heuristic abstract clustering procedures partition a set of objects into clusters such that objects in the same cluster are more similar to each other than objects in different clusters according to some predefined criteria clustering is a popular data analysis and data mining technique since clustering problem have np complete nature the larger the size of the problem the harder to find the optimal solution and furthermore the longer to reach a reasonable results one of the most used techniques for clustering is based on k means such that the data is partitioned into k clusters although k means algorithm is easy to implement and works fast in most situations it suffers from two major drawbacks one is sensitivity to initialization and the other is convergence to local optima it is seen from the studies k harmonic means clustering solves the problem of initialization but since its greedy search nature the second problem convergence to local optima still remains in this paper we develop a new algorithm for solving this problem based on a simulated annealing technique simulated annealing k harmonic means clustering sakhmc the experiment results on the iris and the other well known data illustrate the robustness of the sakhmc clustering algorithm

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments