statistics of shocks in solutions of inviscid burgers equation the purpose of this paper is to analyze statistical properties of discontinuities of solutions of the inviscid burgers equation having a typical realizationb y of the brownian motion as an initial datum this case was proposed and studied numerically in the companion paper by she aurell and frisch the description of the statistics is given in terms of the behavior of the convex hull of the random process w y int limits 0 y b eta eta d eta the hausdorff dimension of the closed set of thosey where the convex hull coincides withw is also studied

pinning and wetting transition for 1 1 dimensional fields with laplacian interaction we consider a random field 1 n as a model for a linear chain attracted to the defect line 0 that is the x axis the free law of the field is specified by the density exp iv i with respect to the lebesgue measure on n where is the discrete laplacian and we allow for a very large class of potentials v the interaction with the defect line is introduced by giving the field a reward 0 each time it touches the x axis we call this model the pinning model we consider a second model the wetting model in which in addition to the pinning reward the field is also constrained to stay nonnegative we show that both models undergo a phase transition as the intensity of the pinning reward varies both in the pinning a p and in the wetting a w case there exists a critical value ca such that when ca the field touches the defect line a positive fraction of times localization while this does not happen for ca delocalization the two critical values are nontrivial and distinct 0 cp cw and they are the only nonanalyticity points of the respective free energies for the pinning model the transition is of second order hence the field at cp is delocalized on the other hand the transition in the wetting model is of first order and for cw the field is localized the core of our approach is a markov renewal theory description of the field

discriminative least squares regression for multiclass classification and feature selection this paper presents a framework of discriminative least squares regression lsr for multiclass classification and feature selection the core idea is to enlarge the distance between different classes under the conceptual framework of lsr first a technique called e dragging is introduced to force the regression targets of different classes moving along opposite directions such that the distances between classes can be enlarged then the e draggings are integrated into the lsr model for multiclass classification our learning framework referred to as discriminative lsr has a compact model form where there is no need to train two class machines that are independent of each other with its compact form this model can be naturally extended for feature selection this goal is achieved in terms of l2 1 norm of matrix generating a sparse learning model for feature selection the model for multiclass classification and its extension for feature selection are finally solved elegantly and efficiently experimental evaluation over a range of benchmark datasets indicates the validity of our method

methods for comparison of coalition influence on games in characteristic function form and their interrelationships abstract this paper extends two existent methods called the blockability relation and the viability relation for simple games to compare influence of coalitions to those for games in characteristic function form and shows that the newly defined relations satisfy transitivity and completeness it is shown in this paper that for every game in characteristic function form the blockability relation and the viability relation have a complementary interrelationship

small values of the maximum for the integral of fractional brownian motion we consider the integral of fractional brownian motion ifbm and its functionals t on the intervals 0 t and t t of the following types the maximum mt the position of the maximum the occupation time above zero etc we show how the asymptotics of p t 1 pt t is related to the hausdorff dimension of lagrangian regular points for the inviscid burgers equation with fbm initial velocity we produce computational evidence in favor of a power asymptotics for pt the data do not reject the hypothesis that the exponent of the power law is related to the similarity parameter h of fractional brownian motion as follows 1 h for the interval t t and h 1 h for 0 t the point 0 is special in that ifbm and its derivative both vanish there

compositional generative mapping for tree structured data part i bottom up probabilistic modeling of trees we introduce a novel compositional recursive probabilistic model for trees that defines an approximated bottom up generative process from the leaves to the root of a tree the proposed model defines contextual state transitions from the joint configuration of the children to the parent nodes we argue that the bottom up context postulates different probabilistic assumptions with respect to a top down approach leading to different representational capabilities we discuss classes of applications that are best suited to a bottom up approach in particular the bottom up context is shown to better correlate and model the co occurrence of substructures among the child subtrees of internal nodes a mixed memory approximation is introduced to factorize the joint children to parent state transition matrix as a mixture of pairwise transitions the proposed approach is the first practical bottom up generative model for tree structured data that maintains the same computational class of its top down counterpart comparative experimental analyses exploiting synthetic and real world datasets show that the proposed model can deal with deep structures better than a top down generative model the model is also shown to better capture structural information from real world data comprising trees with a large out degree the proposed bottom up model can be used as a fundamental building block for the development of other new powerful models

information discriminant analysis feature extraction with an information theoretic objective using elementary information theoretic tools we develop a novel technique for linear transformation from the space of observations into a low dimensional feature subspace for the purpose of classification the technique is based on a numerical optimization of an information theoretic objective function which can be computed analytically the advantages of the proposed method over several other techniques are discussed and the conditions under which the method reduces to linear discriminant analysis are given we show that the novel objective function enjoys many of the properties of the mutual information and the bayes error and we give sufficient conditions for the method to be bayes optimal since the objective function is maximized numerically we show how the calculations can be accelerated to yield feasible solutions the performance of the method compares favorably to other linear discriminant based feature extraction methods on a number of simulated and real world data sets

the multifractal nature of levy processes we show that the sample paths of most levy processes are multifractal functions and we determine their spectrum of singularities

computation of several power indices by generating functions abstract in this paper we propose methods to compute the deegan packel the public good and the shift power indices by generating functions for the particular case of weighted voting games furthermore we define a new power index which combines the ideas of the shift and the deegan packel power indices and also propose a method to compute it with generating functions we conclude by some comments about the complexity to compute these power indices

estimation of the information by an adaptive partitioning of the observation space we demonstrate that it is possible to approximate the mutual information arbitrarily closely in probability by calculating the relative frequencies on appropriate partitions and achieving conditional independence on the rectangles of which the partitions are made empirical results including a comparison with maximum likelihood estimators are presented

persistence of integrated stable processes we compute the persistence exponent of the integral of a stable levy process in terms of its self similarity and positivity parameters this solves a problem raised by shi lower tails of some integrated processes in small deviations and related topics problem panel 2003 along the way we investigate the law of the stable process l evaluated at the first time its integral x hits zero when the bivariate process x l starts from a coordinate axis this extends classical formulae by mckean j math kyoto univ 2 227 235 1963 and gor kov soviet math dokl 16 904 908 1975 for integrated brownian motion

the law of the supremum of a stable levy process with no negative jumps let x xt t 0 be a stable levy process of index fi 2 1 2 with no negative jumps and let st sup0 st xs denote its running supremum for t 0 we show that the density function ft of st can be characterized as the unique solution to a weakly singular volterra integral equation of the flrst kind or equivalently as the unique solution to a flrst order riemann liouville fractional difierential equation satisfying a boundary condition at zero this yields an explicit series representation for ft recalling the familiar relation between st and the flrst entry time x of x into x 1 this further translates into an explicit series representation for the density function of x

estimating optimal feature subsets using efficient estimation of high dimensional mutual information a novel feature selection method using the concept of mutual information mi is proposed in this paper in all mi based feature selection methods effective and efficient estimation of high dimensional mi is crucial in this paper a pruned parzen window estimator and the quadratic mutual information qmi are combined to address this problem the results show that the proposed approach can estimate the mi in an effective and efficient way with this contribution a novel feature selection method is developed to identify the salient features one by one also the appropriate feature subsets for classification can be reliably estimated the proposed methodology is thoroughly tested in four different classification applications in which the number of features ranged from less than 10 to over 15000 the presented results are very promising and corroborate the contribution of the proposed feature selection methodology

input feature selection by mutual information based on parzen window mutual information is a good indicator of relevance between variables and have been used as a measure in several feature selection algorithms however calculating the mutual information is difficult and the performance of a feature selection algorithm depends on the accuracy of the mutual information in this paper we propose a new method of calculating mutual information between input and class variables based on the parzen window and we apply this to a feature selection algorithm for classification problems

compressive projection principal component analysis principal component analysis pca is often central to dimensionality reduction and compression in many applications yet its data dependent nature as a transform computed via expensive eigendecomposition often hinders its use in severely resource constrained settings such as satellite borne sensors a process is presented that effectively shifts the computational burden of pca from the resource constrained encoder to a presumably more capable base station decoder the proposed approach compressive projection pca cppca is driven by projections at the sensor onto lower dimensional subspaces chosen at random while the cppca decoder given only these random projections recovers not only the coefficients associated with the pca transform but also an approximation to the pca transform basis itself an analysis is presented that extends existing rayleigh ritz theory to the special case of highly eccentric distributions this analysis in turn motivates a reconstruction process at the cppca decoder that consists of a novel eigenvector reconstruction based on a convex set optimization driven by ritz vectors within the projected subspaces as such cppca constitutes a fundamental departure from traditional pca in that it permits its excellent dimensionality reduction and compression performance to be realized in an light encoder heavy decoder system architecture in experimental results cppca outperforms a multiple vector variant of compressed sensing for the reconstruction of hyperspectral data

computing banzhaf coleman and shapley shubik power indices with incompatible players in this paper we present methods to compute banzhaf coleman and shapley shubik power indices for weighted majority games when some players are incompatible we use the so called generating functions as a tool

hausdorff dimension of regular points in stochastic burgers flows with levy α stable initial data this paper studies statistical properties of shocks for the inviscid burgers equation with an stable levy motion initial data in the absence of analytic results numerical and computer simulation tools are utilized qualitative and quantitative information on the scaling properties of lagrangian regular points of solutions is obtained and in particular their hausdorff dimension is estimated to be 1 this suggestsa possible extension of ya sinai s result for brownian initial data

maximization of mutual information for supervised linear feature extraction in this paper we present a novel scheme for linear feature extraction in classification the method is based on the maximization of the mutual information mi between the features extracted and the classes the sum of the mi corresponding to each of the features is taken as an heuristic that approximates the mi of the whole output vector then a component by component gradient ascent method is proposed for the maximization of the mi similar to the gradient based entropy optimization used in independent component analysis ica the simulation results show that not only is the method competitive when compared to existing supervised feature extraction methods in all cases studied but it also remarkably outperform them when the data are characterized by strongly nonlinear boundaries between classes

limit clusters in the inviscid burgers turbulence with certain random initial velocities we study the infinite time shock limits given certain markovian initial velocities to the inviscid burgers turbulence specifically we consider the one sided case where initial velocities are zero on the negative half line and follow a time homogeneous nice markov process x on the positive half line finite shock limits occur if the markov process is transient tending to infinity they form a poisson point process if x is spectrally negative we give an explicit description when x is furthermore spatially homogeneous a levy process or a self similar process on 0 we also consider the two sided case where we suppose an independent dual process in the negative spatial direction both spatial homogeneity and an exponential levy condition lead to stationary shock limits

the inviscid burgers equation with initial data of brownian type the solutions to burgers equation in the limit of vanishing viscosity are investigated when the initial velocity is a brownian motion or fractional brownian motion function i e a gaussian process with scaling exponent 0 h 1 typea or the derivative thereof with scaling exponent 1 h 0 typeb largesize numerical experiments are performed helped by the fact that the solution is essentially obtained by performing a legendre transform the main result is obtained for typea and concerns the lagrangian functionx a which gives the location at timet 1 of the fluid particle which started at the locationa it is found to be a complete devil s staircase the cumulative probability of lagrangian shock intervals a also the distribution of shock amplitudes follows a a h law for small a the remaining regular lagrangian locations form a cantor set of dimensionh in eulerian coordinates the shock locations are everywhere dense the scaling properties of various statistical quantities are also found heuristic interpretations are provided for some of these results rigorous results for the case of brownian motion are established in a companion paper by ya sinai for typeb initial velocities e g white noise there are very few small shocks and shock locations appear to be isolated finally it is shown that there are universality classes of random but smooth non scaling initial velocities such that the long time large scale behavior is after rescaling the same as for typea orb

geometric symmetry in the quadratic fisher discriminant operating on image pixels this correspondence examines the design of quadratic fisher discriminants qfds that operate directly on image pixels when image ensembles are taken to comprise all rotated and reflected versions of distinct sample images a procedure based on group theory is devised to identify and discard qfd coefficients made redundant by symmetry for arbitrary sampling lattices this procedure introduces the concept of a degeneracy matrix tensor representations are established for the square lattice point group 8 fold symmetry and hexagonal lattice point group 12 fold symmetry the analysis is largely applicable to the symmetrization of any quadratic filter and generalizes to higher order polynomial volterra filters experiments on square lattice sampled synthetic aperture radar sar imagery verify that symmetrization of qfds can improve their generalization and discrimination ability

subclass discriminant analysis over the years many discriminant analysis da algorithms have been proposed for the study of high dimensional data in a large variety of problems each of these algorithms is tuned to a specific type of data distribution that which best models the problem at hand unfortunately in most problems the form of each class pdf is a priori unknown and the selection of the da algorithm that best fits our data is done over trial and error ideally one would like to have a single formulation which can be used for most distribution types this can be achieved by approximating the underlying distribution of each class with a mixture of gaussians in this approach the major problem to be addressed is that of determining the optimal number of gaussians per class i e the number of subclasses in this paper two criteria able to find the most convenient division of each class into a set of subclasses are derived extensive experimental results are shown using five databases comparisons are given against linear discriminant analysis lda direct lda dlda heteroscedastic lda hlda nonparametric da nda and kernel based lda k lda we show that our method is always the best or comparable to the best

the inviscid burgers equation with brownian initial velocity the law of the hopf cole solution of the inviscid burgers equation with brownian initial velocity is made explicit as examples of applications we investigate the smoothness of the solution the statistical distribution of the shocks we determine the exact hausdorff function of the lagrangian regular points and investigate the existence of lagrangian regular points in a fixed borel set

a characterization of completeness of blockability relations with respect to unanimity this paper gives a characterization of completeness of blockability relations on simple games it is verified in this paper that the blockability relation on a simple game is complete if and only if the simple game is s unanimous for some coalition s in n

holder conditions for the local times and the hausdorff measure of the level sets of gaussian random fields let y t t n be a real valued strongly locally nondeterministic gaussian random field with stationary increments and y 0 0 consider the n d gaussian random field defined by where x 1 x d are independent copies of y the local and global holder conditions in the set variable for the local time of x t are established and the exact hausdorff measure of the level set x 1 x is evaluated

tree structured feature extraction using mutual information one of the most informative measures for feature extraction fe is mutual information mi in terms of mi the optimal fe creates new features that jointly have the largest dependency on the target class however obtaining an accurate estimate of a high dimensional mi as well as optimizing with respect to it is not always easy especially when only small training sets are available in this paper we propose an efficient tree based method for fe in which at each step a new feature is created by selecting and linearly combining two features such that the mi between the new feature and the class is maximized both the selection of the features to be combined and the estimation of the coefficients of the linear transform rely on estimating 2 d mis the estimation of the latter is computationally very efficient and robust the effectiveness of our method is evaluated on several real world data sets the results show that the classification accuracy obtained by the proposed method is higher than that achieved by other fe methods

a method to compare influence of coalitions on group decision other than desirability relation abstract this paper proposes a method to compare influence of coalitions on group decision called blockability relation this relation is defined both on the set of all coalitions and on the set of all feasible coalitions examples show that this relation is definitely different from the desirability relation which have been dealt with in the literature in particular an example shows that the desirability relation is not always transitive and a proposition verifies that the blockability relation is always transitive in order to clarify the simple games on which these relations are defined and to investigate interrelationships among these simple games three types of operators called l intersection l restriction and extension respectively on simple games are defined where l is a family of all feasible coalitions it is verified that the extension of the l intersection of a simple game coincides with the restriction of the simple game

on the hausdorff dimension of regular points of inviscid burgers equation with stable initial data consider an inviscid burgers equation whose initial data is a levy stable process z with 1 we show that when z has positive jumps the hausdorff dimension of the set of lagrangian regular points associated with the equation is strictly smaller than 1 as soon as is close to 1 this gives a partially negative answer to a conjecture of janicki and woyczynski j stat phys 86 1 2 277 299 1997 along the way we contradict a recent conjecture of z shi http www proba jussieu fr pageperso smalldev pbfile pb4 pdf about the lower tails of integrated stable processes

maximum of a fractional brownian motion probabilities of small values let b t b 0 0 be a fractional brownian motion i e a gaussian process with the structure function e b t b s 2 t s 0 2 we study the logarithmic asymptotics of pt p b t 1 t t as t where is either the interval 0 1 or a bounded region that contains a vicinity of 0 for the case of multidimensional time it is shown that ln pt d ln t 1 o 1 where d is the dimension of zeroes of b t in the former case and the dimension of time in the latter

complex random vectors and ica models identifiability uniqueness and separability in this paper the conditions for identifiability separability and uniqueness of linear complex valued independent component analysis ica models are established these results extend the well known conditions for solving real valued ica problems to complex valued models relevant properties of complex random vectors are described in order to extend the darmois skitovich theorem for complex valued models this theorem is used to construct a proof of a theorem for each of the above ica model concepts both circular and noncircular complex random vectors are covered examples clarifying the above concepts are presented

feature fusion using locally linear embedding for classification in most complex classification problems many types of features have been captured or extracted feature fusion is used to combine features for better classification and to reduce data dimensionality kernel based feature fusion methods are very effective for classification but they do not reduce data dimensionality in this brief we propose an effective feature fusion method using locally linear embedding lle the proposed method overcomes the limitations of lle which could not handle different types of features and is inefficient for classification we propose an efficient algorithm to solve the optimization problem in obtaining weights of different features and design an efficient method for lle based classification in comparison to other kernel based feature fusion methods the proposed method fuses features to a significantly lower dimensional feature space with the same discriminant power we have conducted experiments to demonstrate the effectiveness of the proposed feature fusion method

penalized preimage learning in kernel principal component analysis finding the preimage of a feature vector in kernel principal component analysis kpca is of crucial importance when kpca is applied in some applications such as image preprocessing since the exact preimage of a feature vector in the kernel feature space normally does not exist in the input data space an approximate preimage is learned and encouraging results have been reported in the last few years however it is still difficult to find a good estimation of preimage as estimation of preimage in kernel methods is ill posed how to guide the preimage learning for a better estimation is important and still an open problem to address this problem a penalized strategy is developed in this paper where some penalization terms are used to guide the preimage learning process to develop an efficient penalized technique we first propose a two step general framework in which a preimage is directly modeled by weighted combination of the observed samples and the weights are learned by some optimization function subject to certain constraints compared to existing techniques this would also give advantages in directly turning preimage learning into the optimization of the combination weights under this framework a penalized methodology is developed by integrating two types of penalizations first to ensure learning a well defined preimage of which each entry is not out of data range convexity constraint is imposed for learning the combination weights more insight effects of the convexity constraint are also explored second a penalized function is integrated as part of the optimization function to guide the preimage learning process particularly the weakly supervised penalty is proposed discussed and extensively evaluated along with laplacian penalty and ridge penalty it could be further interpreted that the learned preimage can preserve some kind of pointwise conditional mutual information finally kpca with preimage learning is applied on face image data sets in the aspects of facial expression normalization face image denoising recovery of missing parts from occlusion and illumination normalization experimental results show that the proposed preimage learning algorithm obtains lower mean square error mse and better visual quality of reconstructed images