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The hard-sphere model is one of the most extensively studied models in statistical physics. It describes the continuous distribution of spherical particles, governed by hard-core interactions. An important quantity of this model is the normalizing factor of this distribution, called the partition function. We propose a Markov chain Monte Carlo algorithm for approximating the grand-canonical partition function of the hard-sphere model in d dimensions. Up to a fugacity of λ < e/2^d, the runtime of our algorithm is polynomial in the volume of the system. This covers the entire known real-valued regime for the uniqueness of the Gibbs measure. Key to our approach is to define a discretization that closely approximates the partition function of the continuous model. This results in a discrete hard-core instance that is exponential in the size of the initial hard-sphere model. Our approximation bound follows directly from the correlation decay threshold of an infinite regular tree with degree equal to the maximum degree of our discretization. To cope with the exponential blow-up of the discrete instance we use clique dynamics, a Markov chain that was recently introduced in the setting of abstract polymer models. We prove rapid mixing of clique dynamics up to the tree threshold of the univariate hard-core model. This is achieved by relating clique dynamics to block dynamics and adapting the spectral expansion method, which was recently used to bound the mixing time of Glauber dynamics within the same parameter regime., LIPIcs, Vol. 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), pages 66:1-66:15

Logit dynamics is a form of randomized game dynamics where players have a bias towards strategic deviations that give a higher improvement in cost. It is used extensively in practice. In congestion (or potential) games, the dynamics converges to the so-called Gibbs distribution over the set of all strategy profiles, when interpreted as a Markov chain. In general, logit dynamics might converge slowly to the Gibbs distribution, but beyond that, not much is known about their algorithmic aspects, nor that of the Gibbs distribution. In this work, we are interested in the following two questions for congestion games: i) Is there an efficient algorithm for sampling from the Gibbs distribution? ii) If yes, do there also exist natural randomized dynamics that converges quickly to the Gibbs distribution? We first study these questions in extension parallel congestion games, a well-studied special case of symmetric network congestion games. As our main result, we show that there is a simple variation on the logit dynamics (in which we in addition are allowed to randomly interchange the strategies of two players) that converges quickly to the Gibbs distribution in such games. This answers both questions above affirmatively. We also address the first question for the class of so-called capacitated $k$-uniform congestion games. To prove our results, we rely on the recent breakthrough work of Anari, Liu, Oveis-Gharan and Vinzant (2019) concerning the approximate sampling of the base of a matroid according to strongly log-concave probability distribution., Accepted at the 22nd ACM Conference on Economics and Computation (EC 2021)

We introduce a new perfect sampling technique that can be applied to general Gibbs distributions and runs in linear time if the correlation decays faster than the neighborhood growth. In particular, in graphs with sub-exponential neighborhood growth like Z𝑑, our algorithm achieves linear running time as long as Gibbs sampling is rapidly mixing. As concrete applications, we obtain the currently best perfect samplers for colorings and for monomer-dimer models in such graphs.

Markov random fields (MRFs) are well studied during the past 50 years. Their success are mainly due to their flexibility and to the fact that they gives raise to stochastic image models. In this work, we will consider a stochastic differential equation (SDE) driven by Lévy noise. We will show that the solution $ X_v $ of the SDE is a MRF satisfying the Markov property. We will prove that the Gibbs distribution of the process $ X_v $ can be represented graphically through Feynman graphs, which are defined as a set of cliques, then we will provide applications of MRFs in image processing where the image intensity at a particular location depends only on a neighborhood of pixels.

Synthetic aperture radar (SAR) provides rich information about the Earth’s surface under all-weather and day-and-night conditions, and is applied in many relevant fields. SAR imagery semantic segmentation, which can be a final product for end users and a fundamental procedure to support other applications, is one of the most difficult challenges. This paper proposes an encoding-decoding network based on Deeplabv3+ to semantically segment SAR imagery. A new potential energy loss function based on the Gibbs distribution is proposed here to establish the semantic dependence among different categories through the relationship among different cliques in the neighborhood system. This paper introduces an improved channel and spatial attention module to the Mobilenetv2 backbone to improve the recognition accuracy of small object categories in SAR imagery. The experimental results show that the proposed method achieves the highest mean intersection over union (mIoU) and global accuracy (GA) with the least running time, which verifies the effectiveness of our method.

We consider \emph{Gibbs distributions}, which are families of probability distributions over a discrete space $\Omega$ with probability mass function of the form $\mu^\Omega_\beta(\omega) \propto e^{\beta H(\omega)}$ for $\beta$ in an interval $[\beta_{\min}, \beta_{\max}]$ and $H( \omega ) \in \{0 \} \cup [1, n]$. The \emph{partition function} is the normalization factor $Z(\beta)=\sum_{\omega \in\Omega}e^{\beta H(\omega)}$. Two important parameters of these distributions are the log partition ratio $q = \log \tfrac{Z(\beta_{\max})}{Z(\beta_{\min})}$ and the counts $c_x = |H^{-1}(x)|$. These are correlated with system parameters in a number of physical applications and sampling algorithms. Our first main result is to estimate the counts $c_x$ using roughly $\tilde O( \frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and $\tilde O( \frac{n^2}{\varepsilon^2} )$ samples for integer-valued distributions (ignoring some second-order terms and parameters), and we show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings. As a key subroutine, we also develop algorithms to compute the partition function $Z$ using $\tilde O(\frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and using $\tilde O(\frac{n^2}{\varepsilon^2})$ samples for integer-valued distributions., Comment: This is a longer version which extends two previous papers "A Faster Approximation Algorithm for the Gibbs Partition Function" (arXiv:1608.04223), published in COLT 2018, and "Parameter estimation for Gibbs distributions" which will appear in ICALP 2023

Theoretical Computer Science, 745, ISSN:0304-3975

We show that under a low complexity condition on the gradient of a Hamiltonian, Gibbs distributions on the Boolean hypercube are approximate mixtures of product measures whose probability vectors are critical points of an associated mean-field functional. This extends a previous work by the first author. As an application, we demonstrate how this framework helps characterize both Ising models satisfying a mean-field condition and the conditional distributions which arise in the emerging theory of nonlinear large deviations, both in the dense case and in the polynomially-sparse case.

We propose a numerical method to learn Maximum Entropy (MaxEnt) distributions with spatio-temporal constraints from experimental spike trains. This is an extension of two papers [10] and [4] who proposed the estimation of parameters where only spatial constraints were taken into account. The extension we propose allows to properly handle memory effects in spike statistics, for large sized neural networks., Comment: 34 pages, 33 figures

We investigate a probabilistic model for routeing in a multihop ad-hoc communication network, where each user sends a message to the base station. Messages travel in hops via other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution, which favours trajectories with low interference, measured in terms of signal-to-interference ratio. This model was introduced in our earlier paper [KT18], where we expressed, in the limit of a high density of users, the typical distribution of the family of trajectories in terms of a law of large numbers. In the present work, we derive its qualitative properties. We analytically identify the emerging typical scenarios in three extreme regimes. We analyse the typical number of hops and the typical length of a hop, and the deviation of the trajectory from the straight line, (1) in the limit of a large communication area and large distances, and (2) in the limit of a strong interference weight. In both regimes, the typical trajectory approaches a straight line quickly, in regime (1) with equal hop lengths. Interestingly, in regime (1), the typical length of a hop diverges logarithmically in the distance of the transmitter to the base station. We further analyse (3) local and global repulsive effects of a densely populated subarea on the trajectories., Comment: 36 pages, 5 figures

We prove that a stochastic process of pure coagulation has at any time t ≥ 0 a time-dependent Gibbs distribution if and only if the rates ψ(i, j) of single coagulations are of the form ψ(i; j) = if(j) + jf(i), where f is an arbitrary nonnegative function on the set of positive integers. We also obtain a recurrence relation for weights of these Gibbs distributions that allow us to derive the general form of the solution and the explicit solutions in three particular cases of the function f. For the three corresponding models, we study the probability of coagulation into one giant cluster by time t > 0.

We construct an integrated probabilistic model to capture interactions between atoms of a nanocomponent. We then use this model to assess reliabilities of nanocomponents with different structures. Several properties of our proposed model are also described under a sparseness condition. The model is an extension of our previous model based on Markovian random field theory. The proposed integrated model is flexible in that pairwise relationship information among atoms as well as features of individual atoms can be easily incorporated. An important feature that distinguishes the integrated probabilistic model from our previous model is that the integrated approach uses all available sources of information with different weights for different types of interaction. In this paper we consider the nanocomponent at a fixed moment of time, say the present moment, and we assume that the present state of the nanocomponent depends only on the present states of its atoms.

A vast literature has recently been concerned with the analysis of variation in disease counts recorded across geographical areas with the aim of detecting clusters of regions with homogeneous behavior. Most of the proposed modeling approaches have been discussed for the univariate case and only very recently spatial models have been extended to predict more than one outcome simultaneously. In this paper we extend the standard finite mixture models to the analysis of multiple, spatially correlated, counts. Dependence among outcomes is modeled using a set of correlated random effects and estimation is carried out by numerical integration through an EM algorithm without assuming any specific parametric distribution for the random effects. The spatial structure is captured by the use of a Gibbs representation for the prior probabilities of component membership through a Strauss-like model. The proposed model is illustrated using real data.

Using a Gibbs distribution developed in the theory of statistical physics and a long−range percolation theory, we present a new model of a stock price process for explaining the fat tail in the distribution of stock returns. We consider two types of traders, Group A and Group B : Group A traders analyze the past data on the stock market to determine their present trading positions. The way to determine their trading positions is not deterministic but obeys a Gibbs distribution with interactions between the past data and the present trading positions. On the other hand, Group B traders follow the advice reached through the long−range percolation system from the investment adviser. As the resulting stock price process, we derive a Lévy process.

In this paper, we propose a model for image segmentation based on a finite mixture of Gaussian distributions. For each pixel of the image, prior probabilities of class memberships are specified through a Gibbs distribution, where association between labels of adjacent pixels is modeled by a class-specific term allowing for different interaction strengths across classes. We show how model parameters can be estimated in a maximum likelihood framework using Mean Field theory. Experimental performance on perturbed phantom and on real benchmark images shows that the proposed method performs well in a wide variety of empirical situations.

Image modeling using Gibbs priors was shown to be effective in image reconstruction problems. This motivated us to evaluate three techniques for estimating the priors: the heuristic method, the histogram method and Borges' method. We found that both the histogram and Borges' method accurately recovered the parameters needed to specify four different Gibbs distributions from training sets consisting of random sample images from those distributions. This was not the case for the heuristic method. We evaluated the usefulness of the estimated distributions as priors for binary tomography in two experiments: in one the images for the training set were taken from a Gibbs distribution determined by five parameters, and they were typical cardiac phantom images for the second one. We estimated the parameters using both the heuristic and Borges' method in both experiments. We used a modified Metropolis algorithm for the reconstructions. In the first experiment both estimation methods gave rise to excellent reconstructions, but this was not the case for the second experiment; however, even in this case typically less than four percent of the pixels were mis-classified in the reconstructions.

Wavelet threshold algorithms replace small magnitude wavelet coefficients with zero and keep or shrink the other coefficients. This is basically a local procedure, because wavelet coefficients characterize the local regularity of a function. Although a wavelet transform has decorrelating properties, structures in images, like edges, are never decorrelated completely, and these structures appear in the wavelet coefficients: a classification based on a local criterion-like coefficient magnitude is not the perfect method to distinguish important, uncorrupted coefficients from coefficients dominated by noise. We therefore introduce a geometrical prior model for configurations of important wavelet coefficients and combine this with local characterization of a classical threshold procedure into a Bayesian framework. The local characterization is incorporated into the conditional model, whereas the prior model describes only configurations, not coefficient values. More precisely, local characterization favors configurations with clusters of important coefficients. In this way, we can compute, for each coefficient, the posterior probability of being "sufficiently clean." This article proposes and motivates the particular and original choice of the conditional model. Instead of introducing this Bayesian framework, we could also apply heuristic image processing techniques to find clustered configurations of large coefficients. This article also explains the benefits of the Bayesian approach compared to these simple techniques. The parameter of the prior model is estimated on an empirical basis using a pseudolikelihood criterion.

An image restoration can be often formulated as an energy minimization problem. When an energy function is expressed by using the hamiltonian of a classical spin system only with finite range interactions, the probabilistic model, which is described in the form of Gibbs distribution for the energy function, can be regarded as a Markov random field (MRF) model. Some approximate optimization algorithms for the energy minimization problem were proposed in the standpoint of statistical-mechanics. In this paper, the approximate optimization algorithms are summarized and are applied to the image restoration for natural image., 特集 : 「産業におけるソフトコンピューティングに関する国際会議'99」発表論文選集

International audience; This paper presents different recursive formulas for computing the marginals and the normalizing constant of a Gibbs distribution \pi . The common thread is the use of the underlying Markov properties of such processes. The procedures are illustrated with several examples, particularly the Ising model.

We introduce a parametric family for random convex polytopes in Rd which allows for an easy generation of samples for further use, e. g., as random particles in materials modelling and simulation. The basic idea consists in weighting the Poisson cell, which is the typical cell of the stationary and isotropic Poisson hyperplane tessellation, by suitable geometric characteristics. Since this approach results in an exponential family, parameters can be efficiently estimated by maximum likelihood. This work has been motivated by the desire for a flexible model for random convex particles as can be found in many composite materials such as concrete or refractory castables.

In many medical imaging problems such as Positron Emission Tomography (PET), Single Photon Emission Computed Tomography (SPECT) and Transmission Computed Tomography, a lot of projection data for different observation aspects on an object are obtained, allowing reconstruction of the three-dimensional structure of the object from its one and two-dimensional projections. This is basically a problem of analysing incomplete data. Conventional methods such as filtered backprojection (FBP) and Fourier-based methods are deterministic and ignore the statistical elements of the data. Maximum likelihood methods based on suitable statistical models have been found to be satisfactory alternatives to these deterministic methods. The Expectation-Maximisation (EM) algorithm synthesised and presented in a generic form by Dempster, Laird and Rubin in 1977, is an effective iterative method of computing maximum likelihood estimates in a variety of situations, especially in incomplete data problems, where Newton-Raphson types of algorithms are far too cumbersome. In this paper, we review applications of the EM algorithm in tomography and present a bibliography of EM algorithm in tomography.

When analyzing motion observations extracted from image sequences one notes that the histogram of the velocity magnitude at each pixel shows a large probability mass at zero velocity, while the rest of the motion values may be appropriately modeled with a continuous distribution. This suggests the introduction of mixed-state random variables that have probability mass concentrated in discrete states, while they have a probability density over a continuous range of values. In the first part of the chapter, we give a comprehensive description of the theory behind mixed-state statistical models, in particular the development of mixed-state Markov models that permits to take into account spatial and temporal interaction. The presentation generalizes the case of simultaneous modeling of continuous values and any type of discrete symbolic states. For the second part, we present the application of mixed-state models to motion texture analysis. Motion textures correspond to the instantaneous apparent motion maps extracted from dynamic textures. They depict mixed-state motion values with a discrete state at zero and a Gaussian distributionfor the rest. Mixed-state Markov random fields and mixed-state Markov chains are defined and applied to motion texture recognition and tracking. Fil: Crivelli, Tomás. Universidad de Buenos Aires. Facultad de Ingeniería. Departamento de Electronica; Argentina Fil: Bouthemy, Patrick. No especifíca; Fil: Cernuschi Frias, Bruno. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina Fil: Yao, Jian Feng. No especifíca

This paper present recurrence formulas allowing the calculus of the marginals and the normalizing constant of a Gibbs distribution π.The numerical performances of different methods are evaluated on several examples, particularly for an Ising model on a lattice.

International audience; We present a recursive algorithm for the calculation of the marginal of a Gibbs distribution $\pi$. A direct consequence is the calculation of the normalizing constant of$\pi$.

Maximum likelihood parameter estimation is frequently replaced by various techniques because of its intractable normalizing constant. In the same way, the literature displays various alternatives for distributions involving such unreachable constants. In this paper, we consider a Gibbs distribution $\pi $ and present a recurrence formula allowing a recursive calculus of the marginals of $\pi $ and in the same time its normalizing constant$.$ The numerical performance of this algorithm is evaluated for several examples, particularly for an Ising model on a lattice.

We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs-type fragmentation tree with Aldous' beta-splitting model, which has an extended parameter range $\beta>-2$ with respect to the ${\rm beta}(\beta+1,\beta+1)$ probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the two-parameter Poisson--Dirichlet models for exchangeable random partitions of $\mathbb {N}$, with an extended parameter range $0\le\alpha\le1$, $\theta\ge-2\alpha$ and $\alpha, Comment: Published in at http://dx.doi.org/10.3150/08-BEJ134 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

In this article, we propose a novel candidate-gene association test that utilizes a set of tightly linked single nucleotide polymorphisms (SNPs). This is a powerful likelihood ratio test based on Gibbs random field model. We use simulation studies to evaluate the type I error rate of our proposed test, and compare its power with that of other candidate-gene association tests. The simulation results show that our proposed test has correct type I error rate, and is more powerful than the other tests in most cases considered in our simulation studies., 2000 Mathematics Subject Classification: 62P10, 92D10, 92D30, 62F03

The aim of this thesis is the study of statistical properties oflearning algorithm in the case of regression and density estimation.It is divided into three parts.In the first part, the idea is to generalize Olivier Catoni'sPAC-Bayesian theorems about classification to thecase of regression estimation with a general loss function.In the second part, we focus more particularly on the least squareregression and propose a new iterative algorithm for featureselection. This method can be applied to the case of orthonormalfunction basis, leading to optimal rates of convergences, as well asto kernel type functions, leading to some variants of the well-knownSVM method.In the third part, we generalize the method proposed in the secondpart to the density estimation setting with quadratic loss.; Cette thèse a pour objet l'étude despropriétés statistiques d'algorithmes d'apprentissage dans le cas del'estimation de la régression et de la densité. Elle est divisée entrois parties.La première partie consiste en une généralisation des théorèmesPAC-Bayésiens, sur la classification, d'Olivier Catoni, au cas de la régression avec une fonction de pertegénérale.Dans la seconde partie, on étudie plus particulièrement le cas de larégression aux moindres carrés et on propose un nouvel algorithme desélection de variables. Cette méthode peut être appliquée notammentau cas d'une base de fonctions orthonormales, et conduit alors à desvitesses de convergence optimales, mais aussi au cas de fonctions detype noyau, elle conduit alors à une variante des méthodes dites"machines à vecteurs supports" (SVM).La troisième partie étend les résultats de la seconde au cas del'estimation de densité avec perte quadratique.

Molecular dynamic simulations are performed to investigate a long-time evolution of different initial signals in nonlinear acoustic chains with realistic Exp-6 potential and with power potentials. First, in the chain with realistic potential, long-lifetime kink-shaped excitations are found in the system in thermodynamic equilibrium. They give sharp peaks on high-energy tile of Gibbs distribution. Travelling along the chain, each kink collects the energy from the background atoms, and, consequently, transfers it to smallamplitude phonons. Dynamic equilibrium is observed between the processes of growth and decay of the kinks. Проведено числові експерименти за методом молекулярної динаміки з метою дослідження довгочасової еволюції різноманітних початкових сигналів в нелінійнихакустичнихланцюжкахз реалістичним потенціалом виду Ехр-6 та зі ступеневими потенціалами. Впершеулан-цюжку з реалістичним потенціалом знайдені кінкоподібні збудження, що живуть довго, у стані термодинамічної рівноваги. Вони дають гострі піки на високоенергетичному хвості розподілу Гіббса. Прямуючи вздовж ланцюжка кожний кінк збирає енергію з атомів фону, та далі віддає її малоамплітудним фононам. Спостерігається динамічна рівновага між процесами росту та загасання кінків.

It is expected that nano-scale devices and interconnections will introduce unprecedented level of defects in the substrates, and architectural designs need to accommodate the uncertainty inherent at such scales. This consideration motivates the search for new architectural paradigms based on redundancy based defect-tolerant designs. However, redundancy is not always a solution to the reliability problem, and often too much or too little redundancy may cause degradation in reliability. The key challenge is in determining the granularity at which defect tolerance is designed, and the level of redundancy to achieve a specific level of reliability. Analytical probabilistic models to evaluate such reliability-redundancy trade-offs are error prone and cumbersome, and do not scalewell for complex networks of gates. In this thesiswe develop different tools and techniques that can evaluate the reliability measures of combinational circuits, and can be used to analyze reliability-redundancy trade-offs for different defect-tolerant architectural configurations. In particular, we have developed two tools, one of which is based on probabilistic model checking and is named NANOPRISM, and another MATLAB based tool called NANOLAB. We also illustrate the effectiveness of our reliability analysis tools by pointing out certain anomalies which are counter-intuitive but can be easily discovered by these tools, thereby providing better insight into defecttolerant design decisions. We believe that these tools will help furthering research and pedagogical interests in this area, expedite the reliability analysis process and enhance the accuracy of establishing reliability-redundancy trade-off points. Master of Science

Shape and color information are important cues for object recognition. An ideal system should give the option to use both forms of information, as well as the option to use just one of the two. We present in this paper a kernel method that achieves this goal. It is based on results of statistical physics ofd isordered systems combined with Gibbs distributions via kernel functions. Experimental results on a database of 100 objects confirm the effectiveness of the proposed approach.

We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models.

Suppose n particles xi in a region of the plane (possibly representing biological individuals such as trees or smaller organisms) have a joint density proportional to exp{-∑i

Depth from defocus (DFD) problem involves calculating the depth of various points in a scene by modeling the effect that the focal parameters of the camera have on images acquired with a small depth of field. In this paper, we propose a MAP-MRF-based scheme for recovering the depth and the focused image of a scene from two defocused images. The space-variant blur parameter and the focused image of the scene are both modeled as MRFs and their MAP estimates are obtained using simulated annealing. The scheme is amenable to the incorporation of smoothness constraints on the spatial variations of the blur parameter as well as the scene intensity. It also allows for inclusion of line fields to preserve discontinuities. The performance of the proposed scheme is tested on synthetic as well as real data and the estimates of the depth are found to be better than that of the existing window-based DFD technique. The quality of the space-variant restored image of the scene is quite good even under severe space-varying blurring conditions.

This paper presents a reduced-complexity deterministic annealing (DA) approach for vector quantizer (VQ) design by using soft information processing with simplified assignment measures. Low-complexity distributions are designed to mimic the Gibbs distribution, where the latter is the optimal distribution used in the standard DA method. These low-complexity distributions are simple enough to facilitate fast computation, but at the same time they can closely approximate the Gibbs distribution to result in near-optimal performance. We have also derived the theoretical performance loss at a given system entropy due to using the simple soft measures instead of the optimal Gibbs measure. We use thederived result to obtain optimal annealing schedules for the simple soft measures that approximate the annealing schedule for the optimal Gibbs distribution. The proposed reduced-complexity DA algorithms have significantly improved the quality of the final codebooks compared to the generalized Lloyd algorithm and standard stochastic relaxation techniques, both with and without the pairwise nearest neighbor (PNN) codebook initialization. The proposed algorithms are able to evade the local minima and the results show that they are not sensitive to the choice of the initial codebook. Compared to the standard DA approach, the reduced-complexity DA algorithms can operate over 100 times faster with negligible performance difference. For example, for the design of a 16-dimensional vector quantizer having a rate of 0.4375 bit/sample for Gaussian source, the standard DA algorithm achieved 3.60 dB performance in 16 483 CPU seconds, whereas the reduced-complexity DA algorithm achieved the same performance in 136 CPU seconds. Other than VQ design, the DA techniques are applicable to problems such as classification, clustering, and resource allocation.