message passing for maximum weight independent set in this paper we investigate the use of message passing algorithms for the problem of finding the max weight independent set mwis in a graph first we study the performance of the classical loopy max product belief propagation we show that each fixed point estimate of max product can be mapped in a natural way to an extreme point of the linear programming lp polytope associated with the mwis problem however this extreme point may not be the one that maximizes the value of node weights the particular extreme point at final convergence depends on the initialization of max product we then show that if max product is started from the natural initialization of uninformative messages it always solves the correct lp if it converges this result is obtained via a direct analysis of the iterative algorithm and cannot be obtained by looking only at fixed points the tightness of the lp relaxation is thus necessary for max product optimality but it is not sufficient motivated by this observation we show that a simple modification of max product becomes gradient descent on a smoothed version of the dual of the lp and converges to the dual optimum we also develop a message passing algorithm that recovers the primal mwis solution from the output of the descent algorithm we show that the mwis estimate obtained using these two algorithms in conjunction is correct when the graph is bipartite and the mwis is unique finally we show that any problem of maximum a posteriori map estimation for probability distributions over finite domains can be reduced to an mwis problem we believe this reduction will yield new insights and algorithms for map estimation

a monte carlo investigation of the planar lebwohl lasher lattice model abstract a monte carlo computer simulation of a planar version of the lebwohl lasher model is presented the model consists of a set of interaction centres forming a simple square lattice the pair potential is nearest neighbours attractive and varies as a second legendre polynomial of the relative orientation between the two particles five lattice sizes 5 5 10 10 20 20 60 60 and 80 80 of this two dimensional system have been simulated with monte carlo and periodic boundary conditions a study of the orientational pair correlation function indicates a power law decay in the ordered phase and an exponential decay above the pseudo transition temperature our results are consistent with the absence of a true phase transition but also indicate a low temperature phase with long short range order comparisons are made with one existing simulation and with the mean field theory results

phase diagram of onsager crosses onsager crosses are hard nonconvex bodies formed by rigidly connecting three elongated rods equally thick but not necessarily equally long to form perpendicular crosses we study the phase behavior of systems of such particles focusing on their ability to form spatially homogeneous orientationally ordered phases with a symmetry lower than that of the standard uniaxial nematic we treat these systems in the onsager second virial coefficient approximation we apply bifurcation analysis to build up a global picture of the phase diagram which is then refined using approximate numerical calculations finally we generalize the gaussian approximation for the nematic orientational distribution function to deal with the more ordered phases encountered here and compare with the results from the previous techniques to see whether it is feasible to reliably predict the phase diagrams from a computationally cheaper technique

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

statistical mechanics of hard ellipsoids i overlap algorithm and the contact function abstract a contact function for two arbitrary ellipsoids is derived the numerical value of the contact function is less than 1 if they overlap and greater than 1 if they do not this extends previous work by vieillard baron who derived an overlap criterion for spheroids without use of a contact function the equivalence of the two criteria has been checked by extensive numerical test with spheroids it is shown that the use of the contact function greatly facilitates the calculation of the pressure in monte carlo simulations

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

first order transitions for n vector models in two and more dimensions rigorous proof we prove that various so n invariant n vector models with interactions which have a deep and narrow enough minimum have a first order transition in the temperature the result holds in dimensions two or more and is independent of the nature of the low temperature phase

first order phase transition in the two and three dimensional rpn 1 and cpn 1 models in the large n limit solving the lattice rpn 1 and cpn 1 models in the large n limit the authors show that they undergo a first order phase transition for dimensions greater than or equal to two in the latter case although there is no long range order they argue that at least in the rpn 1 model the transition corresponds to a condensation of defects

vector quantization of image subbands a survey subband and wavelet decompositions are powerful tools in image coding because of their decorrelating effects on image pixels the concentration of energy in a few coefficients their multirate multiresolution framework and their frequency splitting which allows for efficient coding matched to the statistics of each frequency band and to the characteristics of the human visual system vector quantization vq provides a means of converting the decomposed signal into bits in a manner that takes advantage of remaining inter and intraband correlation as well as of the more flexible partitions of higher dimensional vector spaces since 1988 a growing body of research has examined the use of vq for subband wavelet transform coefficients we present a survey of these methods

combining fractal image compression and vector quantization in fractal image compression the code is an efficient binary representation of a contractive mapping whose unique fixed point approximates the original image the mapping is typically composed of affine transformations each approximating a block of the image by another block called domain block selected from the same image the search for a suitable domain block is time consuming moreover the rate distortion performance of most fractal image coders is not satisfactory we show how a few fixed vectors designed from a set of training images by a clustering algorithm accelerates the search for the domain blocks and improves both the rate distortion performance and the decoding speed of a pure fractal coder when they are used as a supplementary vector quantization codebook we implemented two quadtree based schemes a fast top down heuristic technique and one optimized with a lagrange multiplier method for the 8 bits per pixel bpp luminance part of the 512 spl kappa 512 lena image our best scheme achieved a peak signal to noise ratio of 32 50 db at 0 25 bpp

design of vector quantizer for image compression using self organizing feature map and surface fitting we propose a new scheme of designing a vector quantizer for image compression first a set of codevectors is generated using the self organizing feature map algorithm then the set of blocks associated with each code vector is modeled by a cubic surface for better perceptual fidelity of the reconstructed images mean removed vectors from a set of training images is used for the construction of a generic codebook further huffman coding of the indices generated by the encoder and the difference coded mean values of the blocks are used to achieve better compression ratio we proposed two indices for quantitative assessment of the psychovisual quality blocking effect of the reconstructed image our experiments on several training and test images demonstrate that the proposed scheme can produce reconstructed images of good quality while achieving compression at low bit rates

belief propagation for the maximum weight independent set and minimum spanning tree problems abstract the belief propagation bp algorithm is a message passing algorithm that is used for solving probabilistic inference problems in practice the bp algorithm performs well as a heuristic in many application fields however the theoretical understanding of bp is limited to improve the theoretical understanding of bp the bp algorithm has been applied to many well understood combinatorial optimization problems in this paper we consider bp applied to the maximum weight independent set mwis and minimum spanning tree mst problems sanghavi et al 2009 12 applied the bp algorithm to the mwis problem we denote their algorithm by bp mwis they showed that if the lp relaxation of the mwis problem has a unique integral optimal solution and bp mwis converges then bp mwis finds the optimal solution also they showed that if the lp relaxation has a non integral optimal solution then bp mwis does not converge in this paper we precisely characterize the graphs for which bp mwis is guaranteed to find the optimal solution regardless of the node weights bayati et al 2008 2 applied the bp algorithm to the mst problem we denote their algorithm by bp mst they showed that if bp mst converges then it finds the optimal solution in this paper however we provide an instance for which bp mst does not converge also since this instance is small and simple we believe that bp mst does not converge for most instances encountered in practice

segmentation based compression for graylevel images abstract this paper presents a segmentation based lossy image compression slic algorithm the segmentation scheme biswas and pal pattern recog lett 21 2000 entropy based and hierarchical in nature provides sub images of homogeneous regions the compression algorithm encodes a graylevel image through global approximations of sub images by 2 d bezier bernstein polynomial along with corrections if needed over regions in sub images by local approximation contours by 1 d bezier bernstein polynomial and texture if present by huffman coding scheme using hilbert scan on texture blocks order of the 2 d polynomials has been computed with the help of an image quality index iqi the proposed compression algorithm also examines the compression result by encoding contours through their approximation based on stretching of discrete circular arcs stretching is done by affine transformation compression results in both the cases have been compared with jpeg results attempts have been made to evaluate the quality of reconstructed images through a fidelity vector whose components are different objective measures

predictive vector quantization with intrablock prediction support region a predictive vector quantization pvq structure is proposed where the encoder uses a predictor based on an intrablock support region followed by a modified vector quantizer stage simulation results show that a modification on a previously published pvq system led to an improvement of 1 db in psnr for lenna

first order transitions for some generalized xy models in this letter we demonstrate the occurrence of first order transitions in temperature for some recently introduced generalized xy models and also point out the connection between them and annealed site diluted lattice gas continuous spin models

conditional entropy constrained residual vq with application to image coding this paper introduces an extension of entropy constrained residual vector quantization vq where intervector dependencies are exploited the method which we call conditional entropy constrained residual vq employs a high order entropy conditioning strategy that captures local information in the neighboring vectors when applied to coding images the proposed method is shown to achieve better rate distortion performance than that of entropy constrained residual vector quantization with less computational complexity and lower memory requirements moreover it can be designed to support progressive transmission in a natural way it is also shown to outperform some of the best predictive and finite state vq techniques reported in the literature this is due partly to the joint optimization between the residual vector quantizer and a high order conditional entropy coder as well as the efficiency of the multistage residual vq structure and the dynamic nature of the prediction

max product for maximum weight matching convergence correctness and lp duality max product belief propagation bp is an iterative message passing algorithm for finding the maximum a posteriori map assignment of a discrete probability distribution specified by a graphical model despite the spectacular success of the algorithm in many application areas such as iterative decoding and combinatorial optimization which involve graphs with many cycles theoretical results about both the correctness and convergence of the algorithm are known in only a few cases see section i for references in this paper we prove the correctness and convergence of max product for finding the maximum weight matching mwm in bipartite graphs even though the underlying graph of the mwm problem has many cycles somewhat surprisingly we show that the max product algorithm converges to the correct mwm as long as the mwm is unique we provide a bound on the number of iterations required and show that for a graph of size n the computational cost of the algorithm scales as o n3 which is the same as the computational cost of the best known algorithms for finding the mwm we also provide an interesting relation between the dynamics of the max product algorithm and the auction algorithm which is a well known distributed algorithm for solving the mwm problem

topological transitions in two dimensional lattice models of liquid crystals the phase diagram of hard core nematogenic models in three dimensional space can be studied by means of onsager s theory and on the other hand the critical properties of continuous interaction potentials can be investigated using the molecular field approach pioneered by maier and saupe comparison between these treatments shows a certain formal similarity reflecting their common variational root on this basis hard core potential models can be mapped onto continuous ones via their excluded volume some years ago this line of reasoning had been applied to hard spherocylinders hence the continuous potential g ensuremath tau a b sqrt 1 ensuremath ensuremath tau 2 bg0 had been used to define a mesogenic model on a three dimensional lattice s romano int j mod phys b 9 85 1995 in the formula ensuremath tau denotes the scalar product between the two unit vectors defining particle orientations here we went on by addressing the same interaction potential on a two dimensional lattice our analysis based on extensive monte carlo simulations found evidence of a topological transition and the critical behavior in its vicinity was studied in detail results obtained for the present model were compared with those already obtained in the literature for interaction potentials defined by legendre polynomials of second and fourth orders in the scalar product ensuremath tau

gradient algorithms for designing predictive vector quantizers a predictive vector quantizer pvq is a vector extension of a predictive quantizer it consists of two parts a conventional memoryless vector quantizer vq and a vector predictor two gradient algorithms for designing a pvq are developed in this paper the steepest descent sd algorithm and the stochastic gradient sg algorithm both have the property of improving the quantizer and the predictor in the sense of minimizing the distortion as measured by the average mean squared error the differences between the two design approaches are the period and the step size used in each iteration to update the codebook and predictor the sg algorithm updates once for each input training vector and uses a small step size while the sd updates only once for a long period possibly one pass over the entire training sequence and uses a relatively large step size code designs and tests are simulated for both gauss markov sources and for sampled speech waveforms and the results are compared to codes designed using techniques that attempt to optimize only the quantizer for the predictor and not vice versa

belief propagation and lp relaxation for weighted matching in general graphs loopy belief propagation has been employed in a wide variety of applications with great empirical success but it comes with few theoretical guarantees in this paper we analyze the performance of the max product form of belief propagation for the weighted matching problem on general graphs we show that the performance of max product is exactly characterized by the natural linear programming lp relaxation of the problem in particular we first show that if the lp relaxation has no fractional optima then max product always converges to the correct answer this establishes the extension of the recent result by bayati shah and sharma which considered bipartite graphs to general graphs perhaps more interestingly we also establish a tight converse namely that the presence of any fractional lp optimum implies that max product will fail to yield useful estimates on some of the edges we extend our results to the weighted b matching and r edge cover problems we also demonstrate how to simplify the max product message update equations for weighted matching making it easily deployable in distributed settings like wireless or sensor networks

phase transition in a nematic n vector model the large n limit the free energy of the rpn 1 model is rigorously computed in the n infinity limit the system shows a first order transition with latent heat in two and three dimensions the first correction in 1 n is also computed comparison of the results with numerical ones show that whereas in three dimensions the large n limit correctly reproduces the behaviour of the system for n or 3 it does not in two dimensions for some observables

image coding using vector quantization a review a review of vector quantization techniques used for encoding digital images is presented first the concept of vector quantization is introduced then its application to digital images is explained spatial predictive transform hybrid binary and subband vector quantizers are reviewed the emphasis is on the usefulness of the vector quantization when it is combined with conventional image coding techniques or when it is used in different domains

belief propagation an asymptotically optimal algorithm for the random assignment problem the random assignment problem asks for the minimum cost perfect matching in the complete n n bipartite graph knn with i i d edge weights say uniform on 0 1 in a remarkable work by aldous aldous d 2001 the 2 limit in the random assignment problem rsa18 381 418 the optimal cost was shown to converge to 2 as n as conjectured by mezard and parisi mezard m g parisi 1987 on the solution of the random link matching problem j phys 48 1451 1459 through the so called cavity method the latter also suggested a nonrigorous decentralized strategy for finding the optimum which turned out to be an instance of the belief propagation bp heuristic discussed by pearl pearl j 1988 probabilistic reasoning in intelligent systems networks of plausible inference morgan kaufmann san francisco in this paper we use the objective method to analyze the performance of bp as the size of the underlying graph becomes large specifically we establish that the dynamic of bp on knn converges in distribution as n to an appropriately defined dynamic on the poisson weighted infinite tree and we then prove correlation decay for this limiting dynamic as a consequence we obtain that bp finds an asymptotically correct assignment in o n2 time only this contrasts with both the worst case upper bound for convergence of bp derived by bayati et al bayati m d shah m sharma 2008 max product for maximum weight matching convergence correctness and lp duality ieee trans inform theory54 3 1241 1251 and the best known computational cost of n3 achieved by edmonds and karp s algorithm edmonds j r karp 1972 theoretical improvements in algorithmic efficiency for network flow problems j acm19 248 264