premixed flame shapes and polynomials abstract the nonlinear nonlocal michelson sivashinsky equation for isolated crests of unstable flames is studied using pole decompositions as starting point polynomials encoding the numerically computed 2 n flame slope poles and auxiliary ones are found to closely follow a meixner pollaczek recurrence accurate steady crest shapes ensue for n 3 squeezed crests ruled by a discretized burgers equation involve the same polynomials such explicit approximate shapes still lack for finite n pole decomposed periodic flames despite another empirical recurrence

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

decomposable graphical gaussian model determination we propose a methodology for bayesian model determination in decomposable graphical gaussian models to achieve this aim we consider a hyper inverse wishart prior distribution on the concentration matrix for each given graph to ensure compatibility across models such prior distributions are obtained by marginalisation from the prior conditional on the complete graph we explore alternative structures for the hyperparameters of the latter and their consequences for the model model determination is carried out by implementing a reversible jump markov chain monte carlo sampler in particular the dimension changing move we propose involves adding or dropping an edge from the graph we characterise the set of moves which preserve the decomposability of the graph giving a fast algorithm for maintaining the junction tree representation of the graph at each sweep as state variable we use the incomplete variance covariance matrix containing only the elements for which the corresponding element of the inverse is nonzero this allows all computations to be performed locally at the clique level which is a clear advantage for the analysis of large and complex datasets finally the statistical and computational performance of the procedure is illustrated by mean of both artificial and real datasets

local marchenko pastur law at the hard edge of sample covariance matrices let xn be a n n matrix whose entries are independent identically distributed complex random variables with mean zero and variance 1n we study the asymptotic spectral distribution of the eigenvalues of the covariance matrix xn xn for n we prove that the empirical density of eigenvalues in an interval e e converges to the marchenko pastur law locally on the optimal scale n e logn b and in any interval up to the hard edge logn bn2 e 4 for any 0 as a consequence we show the complete delocalization of the eigenvectors

polynomials and operator orderings it is shown that there is an exact one to one correspondence between all possible sets of polynomials both orthogonal and nonorthogonal and rules for operator orderings operator orderings that are hermitian give polynomials with definite parity most of the standard classical orthogonal polynomials are associated with operator orderings that are not particularly simple however there is a special one parameter class of hermitian operator orderings that corresponds to a class of elegant but little known orthogonal polynomials called continuous hahn polynomials

covariance matrix selection and estimation via penalised normal likelihood summary we propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix we reparameterise a covariance matrix through the modified cholesky decomposition of its inverse or the one step ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models the cholesky factor containing these regression coefficients is likely to have many off diagonal elements that are zero or close to zero penalised normal likelihoods in this situation with l1 and l2 penalities are shown to be closely related to tibshirani s 1996 lasso approach and to ridge regression adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the cholesky factor while because of its singularity the l1 penalty will set some elements to zero and produce interpretable models an algorithm is developed for computing the estimator and selecting the tuning parameter the proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 x 102 covariance matrix

universality of local eigenvalue statistics for some sample covariance matrices we consider random complex sample covariance matrices 1 x x where x is a p n random matrix with i i d entries of distribution it has been conjectured that both the distribution of the distance between nearest neighbor eigenva lues in the bulk and that of the smallest eigenvalues become in the limit n p n 1 the same as that identified for a complex gaussian distribution we prove these conjectures for a certain class of probability distributions c 2004 wiley periodicals inc

a recursive method for structural learning of directed acyclic graphs in this paper we propose a recursive method for structural learning of directed acyclic graphs dags in which a problem of structural learning for a large dag is first decomposed into two problems of structural learning for two small vertex subsets each of which is then decomposed recursively into two problems of smaller subsets until none subset can be decomposed further in our approach search for separators of a pair of variables in a large dag is localized to small subsets and thus the approach can improve the efficiency of searches and the power of statistical tests for structural learning we show how the recent advances in the learning of undirected graphical models can be employed to facilitate the decomposition simulations are given to demonstrate the performance of the proposed method

estimating high dimensional directed acyclic graphs with the pc algorithm we consider the pc algorithm spirtes et al 2000 for estimating the skeleton and equivalence class of a very high dimensional directed acyclic graph dag with corresponding gaussian distribution the pc algorithm is computationally feasible and often very fast for sparse problems with many nodes variables and it has the attractive property to automatically achieve high computational efficiency as a function of sparseness of the true underlying dag we prove uniform consistency of the algorithm for very high dimensional sparse dags where the number of nodes is allowed to quickly grow with sample size n as fast as o na for any 0 a the sparseness assumption is rather minimal requiring only that the neighborhoods in the dag are of lower order than sample size n we also demonstrate the pc algorithm for simulated data

meixner pollaczek polynomials and the heisenberg algebra an alternative proof is given for the connection between a system of continuous hahn polynomials and identities for symmetric elements in the heisenberg algebra which was first observed by bender mead and pinsky phys rev lett 56 2445 1986 j math phys 28 509 1987 the continuous hahn polynomials turn out to be meixner pollaczek polynomials use is made of the connection between laguerre polynomials and meixner pollaczek polynomials the rodrigues formula for laguerre polynomials an operational formula involving meixner pollaczek polynomials and the schrodinger model for the irreducible unitary representations of the three dimensional heisenberg group

high dimensional gaussian graphical model selection walk summability and local separation criterion we consider the problem of high dimensional gaussian graphical model selection we identify a set of graphs for which an efficient estimation algorithm exists and this algorithm is based on thresholding of empirical conditional covariances under a set of transparent conditions we establish structural consistency or sparsistency for the proposed algorithm when the number of samples n jmin 2 log p where p is the number of variables and jmin is the minimum absolute edge potential of the graphical model the sufficient conditions for sparsistency are based on the notion of walk summability of the model and the presence of sparse local vertex separators in the underlying graph we also derive novel non asymptotic necessary conditions on the number of samples required for sparsistency

a junction tree framework for undirected graphical model selection an undirected graphical model is a joint probability distribution defined on an undirected graph g where the vertices in the graph index a collection of random variables and the edges encode conditional independence relationships among random variables the undirected graphical model selection ugms problem is to estimate the graph g given observations drawn from the undirected graphical model this paper proposes a framework for decomposing the ugms problem into multiple subproblems over clusters and subsets of the separators in a junction tree the junction tree is constructed using a graph that contains a superset of the edges in g we highlight three main properties of using junction trees for ugms first different regularization parameters or different ugms algorithms can be used to learn different parts of the graph this is possible since the subproblems we identify can be solved independently of each other second under certain conditions a junction tree based ugms algorithm can produce consistent results with fewer observations than the usual requirements of existing algorithms third both our theoretical and experimental results show that the junction tree framework does a significantly better job at finding the weakest edges in a graph than existing methods this property is a consequence of both the first and second properties finally we note that our framework is independent of the choice of the ugms algorithm and can be used as a wrapper around standard ugms algorithms for more accurate graph estimation

sample mean versus sample frechet mean for combining complex wishart matrices a statistical study the space of covariance matrices is a non euclidean space the matrices form a manifold which if equipped with a riemannian metric becomes a riemannian manifold and recently this idea has been used for comparison and clustering of complex valued spectral matrices which at a given frequency are typically modeled as complex wishart distributed random matrices identically distributed sample complex wishart matrices can be combined via a standard sample mean to derive a more stable overall estimator however using the riemannian geometry their so called sample frechet mean can also be found we derive the expected value of the determinant of the sample frechet mean and the expected value of the sample frechet mean itself the population frechet mean is shown to be a scaled version of the true covariance matrix the risk under convex loss functions for the standard sample mean is never larger than for the frechet mean in simulations the sample mean also performs better for the estimation of an important functional derived from the estimated covariance matrix namely partial coherence

high dimensional semiparametric bigraphical models in multivariate analysis a gaussian bigraphical model is commonly used for modelling matrix valued data in this paper we propose a semiparametric extension of the gaussian bigraphical model called the nonparanormal bigraphical model a projected nonparametric rank based regularization approach is employed to estimate sparse precision matrices and produce graphs under a penalized likelihood framework theoretically our semiparametric procedure achieves the parametric rates of convergence for both matrix estimation and graph recovery empirically our approach outperforms the parametric gaussian model for non gaussian data and is competitive with its parametric counterpart for gaussian data extensions to the categorical bigraphical model and the missing data problem are discussed copyright 2013 oxford university press

operator norm consistent estimation of large dimensional sparse covariance matrices estimating covariance matrices is a problem of fundamental importance in multivariate statistics in practice it is increasingly frequent to work with data matrices x of dimension n x p where p and n are both large results from random matrix theory show very clearly that in this setting standard estimators like the sample covariance matrix perform in general very poorly in this large n large p setting it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0 and hence this matrix is sparse we develop an estimator to handle this situation the estimator is shown to be consistent in operator norm when for instance we have p n as n oo in other words the largest singular value of the difference between the estimator and the population covariance matrix goes to zero this implies consistency of all the eigenvalues and consistency of eigenspaces associated to isolated eigenvalues we also propose a notion of sparsity for matrices that is compatible with spectral analysis and is independent of the ordering of the variables

a kernel two sample test we propose a framework for analyzing and comparing distributions which we use to construct statistical tests to determine if two samples are drawn from different distributions our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel hilbert space rkhs and is called the maximum mean discrepancy mmd we present two distribution free tests based on large deviation bounds for the mmd and a third test based on the asymptotic distribution of this statistic the mmd can be computed in quadratic time although efficient linear time approximations are available our statistic is an instance of an integral probability metric and various classical metrics on distributions are obtained when alternative function classes are used in place of an rkhs we apply our two sample tests to a variety of problems including attribute matching for databases using the hungarian marriage method where they perform strongly excellent performance is also obtained when comparing distributions over graphs for which these are the first such tests

a connection between operator orderings and representations of the lie algebra mathfrak sl _2 we investigate special classes of polynomials in the quantum mechanical position and momentum operators arising from various operator orderings in particular from the so called orderings generalizing well known operator orderings in quantum mechanics such as the weyl ordering the normal ordering etc viewing orderings as maps from the polynomial algebra on the phase space to the weyl algebra generated by the quantum mechanical position and momentum operators we formulate conditions under which these maps intertwine certain naturally defined actions of the lie algebra these conditions arise via certain regularities in coefficients defining the orderings which can nicely be described in terms of some combinatorial objects called here inverted pascal diagrams at the end we establish a connection between radial elements in the weyl algebra and certain polynomials of the number operator expressible in terms of the hypergeometric function this is related to another representation of realized in terms of difference operators

average density of states for hermitian wigner matrices abstract we consider ensembles of n n hermitian wigner matrices whose entries are up to the symmetry constraints independent and identically distributed random variables assuming sufficient regularity for the probability density function of the entries we show that the expectation of the density of states on arbitrarily small intervals converges to the semicircle law as n tends to infinity

ordered products w algebra and two variable definite parity orthogonal polynomials it has been shown that the cartan subalgebra of w algebra is the space of the two variable definite parity polynomials explicit expressions of these polynomials and their basic properties are presented it also has been shown that they carry the infinite dimensional irreducible representation of the su 1 1 algebra having the spectrum bounded from below a realization of this algebra in terms of difference operators is also obtained for particular values of the ordering parameter s they are identified with the classical orthogonal polynomials of a discrete variable such as the meixner meixner pollaczek and askey wilson polynomials with respect to variable s they satisfy a second order eigenvalue equation of hypergeometric type exact scattering states with zero energy for a family of potentials are expressed in terms of these polynomials it has been put forward that it is the inonu wigner contraction and its inverse that form a bridge between the difference and differential calculus

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

the meixner pollaczek polynomials and a system of orthogonal polynomials in a strip considered in this paper are two systems of polynomials that are orthogonal systems for two different but related inner product spaces one of these systems is a special case of the symmetric meixner pollaczek polynomial systems pn x 2 2 and it turns out that this system is closely related to a system of orthogonal polynomials in the strip s z 1 im z 1 moreover there are some simple operators that connect the systems with each other we have designated the special case of the symmetric meixner pollaczek polynomial systems by n and the latter system on the strip by n and we have been able to show that this system is the limiting case of the symmetric meixner pollaczek polynomial systems pn x 2 2 as 0

joint estimation of multiple graphical models from high dimensional time series in this manuscript we consider the problem of jointly estimating multiple graphical models in high dimensions we assume that the data are collected from n subjects each of which consists of t possibly dependent observations the graphical models of subjects vary but are assumed to change smoothly corresponding to a measure of closeness between subjects we propose a kernel based method for jointly estimating all graphical models theoretically under a double asymptotic framework where both t n and the dimension d can increase we provide the explicit rate of convergence in parameter estimation it characterizes the strength one can borrow across different individuals and the impact of data dependence on parameter estimation empirically experiments on both synthetic and real resting state functional magnetic resonance imaging rs fmri data illustrate the effectiveness of the proposed method

solution of the hausdorff moment problem by the use of pollaczek polynomials abstract we present a new method for solving the hausdorff moment problem which makes use of pollaczek polynomials this problem is ill posed in the sense of hadamard and therefore it is unstable from a numerical viewpoint we shall prove that the expansion which we obtain is asymptotically convergent in the l2 norm when the data i e the moments are perturbed by noise and finite in number this result is largely used in the numerical analysis of the problem

operator orderings and meixner pollaczek polynomials the aim of this paper is to give identities which are generalizations of the formulas given by koornwinder j math phys 30 767 769 1989 10 1063 1 528394 and hamdi zeng j math phys 51 043506 2010 10 1063 1 3372526 our proofs are much simpler than and different from the previous investigations

orthogonal polynomials and operator orderings an alternative and combinatorial proof is given for a connection between a system of hahn polynomials and identities for symmetric elements in the heisenberg algebra which was first observed by bender et al resolution of the operator ordering problem by the method of finite elements phys rev lett 56 2445 1986 continuous hahn polynomials and the heisenberg algebra j math phys 28 509 1987 and proven by koornwinder meixner pollaczek polynomials and the heisenberg algebra j math phys 30 767 1989 in the same vein two results announced by bender and dunne polynomials and operator orderings j math phys 29 1727 1988 connecting a special one parameter class of hermitian operator orderings and the continuous hahn polynomials are also proven

generalized zernike or disc polynomials an application in quantum optics an application of the generalized zernike or disc polynomials recently introduced in the literature is shown resorting to the lie algebra based investigation of the dynamics of quantum systems driven by two mode interaction hamiltonians further properties of the associated disc functions are deduced also a generalization of the disc polynomials towards the hahn polynomials is suggested

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

extended bayesian information criteria for model selection with large model spaces summary the ordinary bayesian information criterion is too liberal for model selection when the model space is large in this paper we re examine the bayesian paradigm for model selection and propose an extended family of bayesian information criteria which take into account both the number of unknown parameters and the complexity of the model space their consistency is established in particular allowing the number of covariates to increase to infinity with the sample size their performance in various situations is evaluated by simulation studies it is demonstrated that the extended bayesian information criteria incur a small loss in the positive selection rate but tightly control the false discovery rate a desirable property in many applications the extended bayesian information criteria are extremely useful for variable selection in problems with a moderate sample size but with a huge number of covariates especially in genome wide association studies which are now an active area in genetics research

integrable systems on the lattice and orthogonal polynomials of discrete variable some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored

the unitary irreducible representations of sl 2 r in all subgroup reductions we use the canonical transform realization of sl 2 r in order to find all matrix elements and integral kernels for the unitary irreducible representations of this group explicit results are given for all mixed bases and subgroup reductions these provide the full multiparameter set of integral transforms and series expansions associated to sl 2 r

on the connection between continuous hahn polynomials and the heisenberg algebra an elementary proof is given for the connection between a system of continuous hahn polynomials and symmetrizations of elements in the heisenberg algebra bender mead and pinsky phys rev lett 56 2445 1986 j math phys 28 509 1987 koornwinder ibid 30 767 1989

latent variable graphical model selection via convex optimization suppose we observe samples of a subset of a collection of random variables no additional information is provided about the number of latent variables nor of the relationship between the latent and observed variables is it possible to discover the number of latent components and to learn a statistical model over the entire collection of variables we address this question in the setting in which the latent and observed variables are jointly gaussian with the conditional statistics of the observed variables conditioned on the latent variables being specified by a graphical model as a first step we give natural conditions under which such latent variable gaussian graphical models are identifiable given marginal statistics of only the observed variables essentially these conditions require that the conditional graphical model among the observed variables is sparse while the effect of the latent variables is spread out over most of the observed variables next we propose a tractable convex program based on regularized maximum likelihood for model selection in this latent variable setting the regularizer uses both the ell 1 norm and the nuclear norm our modeling framework can be viewed as a combination of dimensionality reduction to identify latent variables and graphical modeling to capture remaining statistical structure not attributable to the latent variables and it consistently estimates both the number of latent components and the conditional graphical model structure among the observed variables these results are applicable in the high dimensional setting in which the number of latent observed variables grows with the number of samples of the observed variables the geometric properties of the algebraic varieties of sparse matrices and of low rank matrices play an important role in our analysis

weyl ordered polynomials in fractional dimensional quantum mechanics we develop algebraic properties of weyl ordered polynomials in the momentum and position operators p q which satisfy the r deformed heisenberg algebra representations of which describe quantum mechanics in fractional dimensions by viewing weyl ordered polynomials as tensor operators with respect to the lie algebra we derive a specific form for these polynomials including an expression in terms of hypergeometric functions and determine various algebraic properties such as recurrence relations symmetries and also a general product formula from which all commutators and anti commutators may be calculated we briefly discuss several applications to quantum mechanics in fractional dimensions

blessing of massive scale spatial graphical model estimation with a total cardinality constraint approach we consider the problem of estimating high dimensional spatial graphical models with a total cardinality constraint i e the ell 0 constraint though this problem is highly nonconvex we show that its primal dual gap diminishes linearly with the dimensionality and provide a convex geometry justification of this blessing of massive scale phenomenon motivated by this result we propose an efficient algorithm to solve the dual problem which is concave and prove that the solution achieves optimal statistical properties extensive numerical results are also provided

high dimensional semiparametric gaussian copula graphical models ing the spearman s rho and kendall s tau we prove that the nonparanormal skeptic achieves the optimal parametric rates of convergence for both graph recovery and parameter estimation this result suggests that the nonparanormal graphical models can be used as a safe replacement of the popular gaussian graphical models even when the data are truly gaussian besides theoretical analysis we also conduct thorough numerical simulations to compare the graph recovery performance of dierent estimators under both ideal and noisy settings the proposed methods are then applied on a largescale genomic dataset to illustrate their empirical usefulness the r package huge implementing the proposed methods is available on the comprehensive r archive network http cran r project org

finding non overlapping clusters for generalized inference over graphical models graphical models use graphs to compactly capture stochastic dependencies amongst a collection of random variables inference over graphical models corresponds to finding marginal probability distributions given joint probability distributions in general this is computationally intractable which has led to a quest for finding efficient approximate inference algorithms we propose a framework for generalized inference over graphical models that can be used as a wrapper for improving the estimates of approximate inference algorithms instead of applying an inference algorithm to the original graph we apply the inference algorithm to a block graph defined as a graph in which the nodes are non overlapping clusters of nodes from the original graph this results in marginal estimates of a cluster of nodes which we further marginalize to get the marginal estimates of each node our proposed block graph construction algorithm is simple efficient and motivated by the observation that approximate inference is more accurate on graphs with longer cycles we present extensive numerical simulations that illustrate our block graph framework with a variety of inference algorithms e g those in the libdai software package these simulations show the improvements provided by our framework

continuous hahn polynomials and the heisenberg algebra continuous hahn polynomials have surfaced in a number of somewhat obscure physical applications for example they have emerged in the description of two photon processes in hydrogen hard hexagon statistical mechanical models and clebsch gordan expansions for unitary representations of the lorentz group so 3 1 in this paper it is shown that there is a simple and elegant way to construct these polynomials using the heisenberg algebra

the hahn and meixner polynomials of an imaginary argument and some of their applications the hahn and meixner polynomials belonging to the classical orthogonal polynomials of a discrete variable are analytically continued in the complex plane both in variable and parameter this leads to the origination of two systems of real polynomials orthogonal with respect to a continuous measure the meixner polynomials of an imaginary argument obtained in this manner turned out to be known in the literature as the pollaczek polynomials the orthogonality relation for the hahn polynomials with respect to a continuous measure is apparently new a close connection between the hahn polynomials of an imaginary argument and representations of the lorentz group so 3 1 is considered

structural similarity and difference testing on multiple sparse gaussian graphical models we present a new framework on inferring structural similarities and differences among multiple high dimensional gaussian graphical models ggms corresponding to the same set of variables under distinct experimental conditions the new framework adopts the partial correlation coefficients to characterize the potential changes of dependency strengths between two variables a hierarchical method has been further developed to recover edges with different or similar dependency strengths across multiple ggms in particular we first construct two sample test statistics for testing the equality of partial correlation coefficients and conduct large scale multiple tests to estimate the substructure of differential dependencies after removing differential substructure from original ggms a follow up multiple testing procedure is used to detect the substructure of similar dependencies among ggms in each step false discovery rate is controlled asymptotically at a desired level power results are proved which demonstrate that our method is more powerful on finding common edges than the common approach that separately estimates ggms the performance of the proposed hierarchical method is illustrated on simulated datasets

hausdorff moments hardy spaces and power series in this paper we consider power and trigonometric series whose coefficients are supposed to satisfy the hausdorff conditions which play a relevant role in the moment problem theory we prove that these series converge to functions analytic in cut domains we are then able to reconstruct the jump functions across the cuts from the coefficients of the series expansions by the use of the pollaczek polynomials we can thus furnish a solution for a class of cauchy integral equations

underdetermined systems of partial differential equations this paper examines underdetermined systems of partial differential equations in which the independent variables may be classical c numbers or even quantum operators one can view an underdetermined system as expressing the kinematic constraints on a set of dynamical variables that generate a lie algebra the arbitrariness in the general solution reflects the freedom to specify the dynamics of such a system

heisenberg weyl algebra revisited combinatorics of words and paths the heisenberg weyl algebra which underlies virtually all physical representations of quantum theory is considered from the combinatorial point of view we provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules we also discuss the rook problem on the associated ferrers board this is related to the calculus in the normally ordered basis from this starting point we explore combinatorial underpinning of the heisenberg weyl algebra which offers novel perspectives methods and applications

high dimensional ising model selection using l1 regularized logistic regression we consider the problem of estimating the graph associated with a binary ising markov random field we describe a method based on l1 regularized logistic regression in which the neighborhood of any given node is estimated by performing logistic regression subject to an l1 constraint the method is analyzed under high dimensional scaling in which both the number of nodes p and maximum neighborhood size d are allowed to grow as a function of the number of observations n our main results provide sufficient conditions on the triple n p d and the model parameters for the method to succeed in consistently estimating the neighborhood of every node in the graph simultaneously with coherence conditions imposed on the population fisher information matrix we prove that consistent neighborhood selection can be obtained for sample sizes n d3log p with exponentially decaying error when these same conditions are imposed directly on the sample matrices we show that a reduced sample size of n d2log p suffices for the method to estimate neighborhoods consistently although this paper focuses on the binary graphical models we indicate how a generalization of the method of the paper would apply to general discrete markov random fields

model selection and estimation in the gaussian graphical model summary we propose penalized likelihood methods for estimating the concentration matrix in the gaussian graphical model the methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite and thus conduct model selection and estimation simultaneously the implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization we propose a bic type criterion for the selection of the tuning parameter in the penalized likelihood methods the connection between our methods and existing methods is illustrated simulations and real examples demonstrate the competitive performance of the new methods

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

optimal rates of convergence for covariance matrix estimation covariance matrix plays a central role in multivariate statistical analysis significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices however a minimax theory has yet been developed in this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and frobenius norm it is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation the minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties a key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound the technical analysis requires new ideas that are quite different from those used in the more conventional function sequence estimation problems

walk sums and belief propagation in gaussian graphical models we present a new framework based on walks in a graph for analysis and inference in gaussian graphical models the key idea is to decompose the correlation between each pair of variables as a sum over all walks between those variables in the graph the weight of each walk is given by a product of edgewise partial correlation coefficients this representation holds for a large class of gaussian graphical models which we call walk summable we give a precise characterization of this class of models and relate it to other classes including diagonally dominant attractive non frustrated and pairwise normalizable we provide a walk sum interpretation of gaussian belief propagation in trees and of the approximate method of loopy belief propagation in graphs with cycles the walk sum perspective leads to a better understanding of gaussian belief propagation and to stronger results for its convergence in loopy graphs

learning latent tree graphical models we study the problem of learning a latent tree graphical model where samples are available only from a subset of variables we propose two consistent and computationally efficient algorithms for learning minimal latent trees that is trees without any redundant hidden nodes unlike many existing methods the observed nodes or variables are not constrained to be leaf nodes our algorithms can be applied to both discrete and gaussian random variables and our learned models are such that all the observed and latent variables have the same domain state space our first algorithm recursive grouping builds the latent tree recursively by identifying sibling groups using so called information distances one of the main contributions of this work is our second algorithm which we refer to as clgrouping clgrouping starts with a pre processing procedure in which a tree over the observed variables is constructed this global step groups the observed nodes that are likely to be close to each other in the true latent tree thereby guiding subsequent recursive grouping or equivalent procedures such as neighbor joining on much smaller subsets of variables this results in more accurate and efficient learning of latent trees we also present regularized versions of our algorithms that learn latent tree approximations of arbitrary distributions we compare the proposed algorithms to other methods by performing extensive numerical experiments on various latent tree graphical models such as hidden markov models and star graphs in addition we demonstrate the applicability of our methods on real world data sets by modeling the dependency structure of monthly stock returns in the s p index and of the words in the 20 newsgroups data set

a constrained l1 minimization approach to sparse precision matrix estimation this article proposes a constrained l1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p variate random variables the resulting estimator is shown to have a number of desirable properties in particular the rate of convergence between the estimator and the true s sparse precision matrix under the spectral norm is when the population distribution has either exponential type tails or polynomial type tails we present convergence rates under the elementwise l norm and frobenius norm in addition we consider graphical model selection the procedure is easily implemented by linear programming numerical performance of the estimator is investigated using both simulated and real data in particular the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods

the algebra of two dimensional generalized chebyshev koornwinder oscillator in the previous works of borzov and damaskinsky chebyshev koornwinder oscillator theor math phys 175 3 765 772 2013 and ladder operators for chebyshev koornwinder oscillator in proceedings of the days on diffraction 2013 the authors have defined the oscillator like system that is associated with the two variable chebyshev koornwinder polynomials we call this system the generalized chebyshev koornwinder oscillator in this paper we study the properties of infinite dimensional lie algebra that is analogous to the heisenberg algebra for the chebyshev koornwinder oscillator we construct the exact irreducible representation of this algebra in a hilbert space of functions that are defined on a region which is bounded by the steiner hypocycloid the functions are square integrable with respect to the orthogonality measure for the chebyshev koornwinder polynomials and these polynomials form an orthonormalized basis in the space the generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators h h

the symmetric meixner pollaczek polynomials with real parameter abstract the symmetric meixner pollaczek polynomials p n x 2 2 for 0 are well studied polynomials these are polynomials orthogonal on the real line with respect to a continuous positive real measure for 0 p n x 2 2 are also polynomials however they are not orthogonal on the real line with respect to any real measure this paper defines a non standard inner product with respect to which the polynomials p n x 2 2 for 0 become orthogonal polynomials it examines the major properties of the polynomials p n x 2 2 for 0 which are also shared by the polynomials p n x 2 2 for 0

functional graphical models graphical models have attracted increasing attention in recent years especially in settings involving high dimensional data in particular gaussian graphical models are used to model the conditio

transformation properties of the schrodinger equation with the momentum dependent potential x 2ord using suitable coordinate transformations enables us to convert the schrodinger equation with the momentum dependent potential x 2ord into a pair of coupled s wave poschl teller systems with zero energy an equivalent zero energy system with a related singular potential has also been derived this time by using an appropriate wavefunction transformation

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

high dimensional sparse inverse covariance estimation using greedy methods in this paper we consider the task of estimating the non zero pattern of the sparse inverse covariance matrix of a zero mean gaussian random vector from a set of iid samples note that this is also equivalent to recovering the underlying graph structure of a sparse gaussian markov random field gmrf we present two novel greedy approaches to solving this problem the first estimates the non zero covariates of the overall inverse covariance matrix using a series of global forward and backward greedy steps the second estimates the neighborhood of each node in the graph separately again using greedy forward and backward steps and combines the intermediate neighborhoods to form an overall estimate the principal contribution of this paper is a rigorous analysis of the sparsistency of these two greedy procedures that is their consistency in recovering the sparsity pattern of the inverse covariance matrix surprisingly we show that both the local and global greedy methods learn the full structure of the model with high probability given just o d log p samples which is a significant improvement over state of the art 1 regularized gaussian mle graphical lasso that requires o d 2 log p samples moreover the restricted eigenvalue and smoothness conditions imposed by our greedy methods are much weaker than the strong irrepresentable conditions required by the 1 regularization based methods we corroborate our results with extensive simulations and examples comparing our local and

joint estimation of multiple conditional gaussian graphical models in this paper we propose a joint conditional graphical lasso to learn multiple conditional gaussian graphical models also known as gaussian conditional random fields with some similar structures our model builds on the maximum likelihood method with the convex sparse group lasso penalty moreover our model is able to model multiple multivariate linear regressions with unknown noise covariances via a convex formulation in addition we develop an efficient approximated newton s method for optimizing our model theoretically we establish the asymptotic properties of our model on consistency and sparsistency under the high dimensional settings finally extensive numerical results on simulations and real data sets demonstrate that our method outperforms the compared methods on structure recovery and structured output prediction to the best of our knowledge the joint learning of multiple multivariate regressions with unknown covariance is first studied

discrete time quantum mechanics abstract this paper summarizes a research program that has been underway for a decade the objective is to find a fast and accurate scheme for solving quantum problems which does not involve a monte carlo algorithm we use an alternative strategy based on the method of finite elements we are able to formulate fully consistent quantum mechanical systems directly on a lattice in terms of operator difference equations one advantage of this discretized formulation of quantum mechanics is that the ambiguities associated with operator ordering are eliminated furthermore the scheme provides an easy way in which to obtain the energy levels of the theory numerically a generalized version of this discretization scheme can be applied to quantum field theory problems the difficulties normally associated with fermion doubling are eliminated also one can incorporate local gauge invariance in the finite element formulation results for some field theory models are summarized in particular we review the calculation of the anomaly in two dimensional quantum electrodynamics the schwinger model finally we discuss nonabelian gauge theories

gaussian markov distributions over finite graphs on s interesse aux modeles de selection de covariance introduites par dempster 1972 et etudies par wermuth 1976

continuous hahn polynomials a slightly more general orthogonality relation for the hahn polynomials of a continuous variable than the recent one given by atakishiyev and suslov 1985 is given here

nonparametric estimation of large covariance matrices of longitudinal data summary estimation of an unstructured covariance matrix is difficult because of its positivedefiniteness constraint this obstacle is removed by regressing each variable on its predecessors so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying coefficient and varying order regression models our framework is similar to the use of increasing order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series as an illustration we adopt fan zhang s 2000 two step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite for parsimony a suitable order for the sequence of auto regression models is found using penalised likelihood criteria like aic and bic some asymptotic results for the local polynomial estimators of components of a covariance matrix are established two longitudinal datasets are analysed to illustrate the methodology a simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices

the nonparanormal semiparametric estimation of high dimensional undirected graphs recent methods for estimating sparse undirected graphs for real valued data in high dimensional problems rely heavily on the assumption of normality we show how to use a semiparametric gaussian copula or nonparanormal for high dimensional inference just as additive models extend linear models by replacing linear functions with a set of one dimensional smooth functions the nonparanormal extends the normal by transforming the variables by smooth functions we derive a method for estimating the nonparanormal study the method s theoretical properties and show that it works well in many examples

bounds for the stieltjes transform and the density of states of wigner matrices we consider ensembles of wigner matrices whose entries are up to the symmetry constraints independent and identically distributed random variables we show the convergence of the stieltjes transform towards the stieltjes transform of the semicircle law on optimal scales and with the optimal rate our bounds improve previous results in particular from erd s et al adv math 229 3 1435 1515 2012 electron j probab 18 59 1 58 2013 by removing the logarithmic corrections as applications we establish the convergence of the eigenvalue counting functions with the rate log n n and the rigidity of the eigenvalues of wigner matrices on the same scale these bounds improve the results of erd s et al adv math 229 3 1435 1515 2012 electron j probab 18 59 1 58 2013 gotze and tikhomirov 2013

from sequences to polynomials and back via operator orderings bender and dunne polynomials and operator orderings j math phys 29 1727 1731 1988 showed that linear combinations of words qkpnqn k where p and q are subject to the relation qp pq i may be expressed as a polynomial in the symbol z 12 qp pq relations between such polynomials and linear combinations of the transformed coefficients are explored in particular examples yielding orthogonal polynomials are provided

model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data we consider the problem of estimating the parameters of a gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse our approach is to solve a maximum likelihood problem with an added l1 norm penalty term the problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes we present two new algorithms for solving problems with at least a thousand nodes in the gaussian case our first algorithm uses block coordinate descent and can be interpreted as recursive l1 norm penalized regression our second algorithm based on nesterov s first order method yields a complexity estimate with a better dependence on problem size than existing interior point methods using a log determinant relaxation of the log partition function wainwright and jordan 2006 we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case we test our algorithms on synthetic data as well as on gene expression and senate voting records data

universality of the local spacing distribution in certain ensembles of hermitian wigner matrices consider an n n hermitian random matrix with independent entries not necessarily gaussian a so called wigner matrix it has been conjectured that the local spacing distribution i e the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is in the limit as n the same as that of hermitian random matrices from gue we prove this conjecture for a certain subclass of hermitian wigner matrices