poisson approximations for the ising model a d dimensional ising model on a lattice torus is considered as the size n of the lattice tends to infinity a poisson approximation is given for the distribution of the number of copies in the lattice of any given local configuration provided the magnetic field a a n tends to and the pair potential b remains fixed using the stein chen method a bound is given for the total variation error in the ferromagnetic case

cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

optimal detection of sparse principal components in high dimension we perform a finite sample analysis of the detection levels for sparse principal components of a high dimensional covariance matrix our minimax optimal test is based on a sparse eigenvalue statistic alas computing this test is known to be np complete in general and we describe a computationally efficient alternative test using convex relaxations our relaxation is also proved to detect sparse principal components at near optimal detection levels and performs very well on simulated datasets ams 2000 subject classifications primary 62h25 secondary 62f04 90c22

fast and stable subspace tracking we consider the problem of adaptive subspace tracking when the rank of the subspace we seek to estimate is assumed to be known starting from the data projection method dpm which constitutes a simple and reliable means for adaptively estimating and tracking subspaces we develop a fast and numerically robust implementation of dpm which comes at a considerably lower computational cost most existing schemes track subspaces corresponding either to the largest or to the smallest singular values while our dpm version can switch from one subspace type to the other with a simple change of sign of its single parameter the proposed algorithm provides orthonormal vector estimates of the subspace basis that are numerically stable since they do not accumulate roundoff errors in fact our scheme constitutes the first numerically stable low complexity algorithm for tracking subspaces corresponding to the smallest singular values regarding convergence towards orthonormality our scheme exhibits the fastest speed among all other subspace tracking algorithms of similar complexity

on consistency and sparsity for principal components analysis in high dimensions principal components analysis pca is a classic method for the reduction of dimensionality of data in the form of n observations or cases of a vector with p variables contemporary datasets often have p comparable with or even much larger than n our main assertions in such settings are a that some initial reduction in dimensionality is desirable before applying any pca type search for principal modes and b the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation we describe a simple asymptotic model in which the estimate of the leading principal component vector via standard pca is consistent if and only if p n n 0 we provide a simple algorithm for selecting a subset of coordinates with largest sample variances and show that if pca is done on the selected subset then consistency is recovered even if p n n

adaptive rank estimation for spherical subspace trackers we develop a rank tracking method for spherical subspace trackers the method adaptively sets a threshold based on the averaged noise eigenvalue the signal eigenvalues are then compared with the threshold to reach a decision on the subspace rank the threshold itself is chosen to balance the error probabilities due to rank underfitting and overfitting simulation results show that our method performs as well as and for very low snrs even better than information theoretic criteria

finite cycle gibbs measures on permutations of mathbb z d we consider gibbs distributions on the set of permutations of mathbb z d associated to the hamiltonian h sigma sum x v sigma x x where sigma is a permutation and v mathbb z d rightarrow mathbb r is a strictly convex potential call finite cycle those permutations composed by finite cycles only we give conditions on v ensuring that for large enough temperature alpha 0 there exists a unique infinite volume ergodic gibbs measure mu alpha concentrating mass on finite cycle permutations this measure is equal to the thermodynamic limit of the specifications with identity boundary conditions we construct mu alpha as the unique invariant measure of a markov process on the set of finite cycle permutations that can be seen as a loss network a continuous time birth and death process of cycles interacting by exclusion an approach proposed by fernandez ferrari and garcia define tau v as the shift permutation tau v x x v in the gaussian case v vert cdot vert 2 we show that for each v in mathbb z d mu alpha v given by mu alpha v f mu alpha f tau v cdot is an ergodic gibbs measure equal to the thermodynamic limit of the specifications with tau v boundary conditions for a general potential v we prove the existence of gibbs measures mu alpha v when alpha is bigger than some v dependent value

non homogeneous polygonal markov fields in the plane graphical representations and geometry of higher order correlations we consider polygonal markov fields originally introduced by arak and surgailis probab theory relat fields 80 543 579 1989 our attention is focused on fields with nodes of order two which can be regarded as continuum ensembles of non intersecting contours in the plane sharing a number of features with the two dimensional ising model we introduce non homogeneous version of polygonal fields in anisotropic environment for these fields we provide a class of new graphical constructions and random dynamics these include a generalized dynamic representation generalized and defective disagreement loop dynamics as well as a generalized contour birth and death dynamics next we use these constructions as tools to obtain new exact results on the geometry of higher order correlations of polygonal markov fields in their consistent regime

hitting times for independent random walks on ℤd we consider a system of asymmetric independent random walks on z d denoted by t t e r stationary under the product poisson measure v p of marginal density p 0 we fix a pattern a an increasing local event and denote by the hitting time of a by using a loss network representation of our system at small density we obtain a coupling between the laws of t conditioned on r t for all times t when d 3 this provides bounds on the rate of convergence of the law of t conditioned on r t toward its limiting probability measure as t tends to infinity we also treat the case where the initial measure is close to v without being product

condition number regularized covariance estimation estimation of high dimensional covariance matrices is known to be a difficult problem has many applications and is of current interest to the larger statistics community in many applications including so called the large p small n setting the estimate of the covariance matrix is required to be not only invertible but also well conditioned although many regularization schemes attempt to do this none of them address the ill conditioning problem directly in this paper we propose a maximum likelihood approach with the direct goal of obtaining a well conditioned estimator no sparsity assumption on either the covariance matrix or its inverse are are imposed thus making our procedure more widely applicable we demonstrate that the proposed regularization scheme is computationally efficient yields a type of steinian shrinkage estimator and has a natural bayesian interpretation we investigate the theoretical properties of the regularized covariance estimator comprehensively including its regularization path and proceed to develop an approach that adaptively determines the level of regularization that is required finally we demonstrate the performance of the regularized estimator in decision theoretic comparisons and in the financial portfolio optimization setting the proposed approach has desirable properties and can serve as a competitive procedure especially when the sample size is small and when a well conditioned estimator is required

hub discovery in partial correlation graphs one of the most important problems in large scale inference problems is the identification of variables that are highly dependent on several other variables when dependence is measured by partial correlations these variables identify those rows of the partial correlation matrix that have several entries with large magnitudes i e hubs in the associated partial correlation graph this paper develops theory and algorithms for discovering such hubs from a few observations of these variables we introduce a hub screening framework in which the user specifies both a minimum partial correlation rho and a minimum degree delta to screen the vertices the choice of rho and delta can be guided by our mathematical expressions for the phase transition correlation threshold rho c governing the average number of discoveries they can also be guided by our asymptotic expressions for familywise discovery rates under the assumption of large number p of variables fixed number n of multivariate samples and weak dependence under the null hypothesis that the dispersion covariance matrix is sparse these limiting expressions can be used to enforce familywise error constraints and to rank the discoveries in order of increasing statistical significance for n ll p the computational complexity of the proposed partial correlation screening method is low and is therefore highly scalable thus it can be applied to significantly larger problems than previous approaches the theory is applied to discovering hubs in a high dimensional gene microarray dataset

an algebraic approach to the classification of centers in polynomial lienard systems abstract we consider the second order lienard system x f x x g x 0 where f x and g x are polynomials which we rewrite in the equivalent two dimensional form x y y g x yf x we show that the local question of whether a critical point of this system is a center can be expressed in terms of global conditions on f and g using these results we give a simple classification of all such centers we also address the problem of the coexistence of centers foci and limit cycles for these systems systems with degenerate centers are also considered

fast rank adaptive subspace tracking and applications high computational complexity and inadequate parallelism have deterred the use of subspace based algorithms in real time systems we proposed a new class of fast subspace tracking fst algorithms that overcome these problems by exploiting the matrix structure inherent in multisensor processing these algorithms simultaneously track an orthonormal basis for the signal subspace and preserve signal eigenstructure information while requiring only o nr operations per update where n is the number of channels and r is the effective rank because of their low computational complexity these algorithms have applications in both recursive and block data processing because they preserve the signal eigenstructure as well as compute an orthonormal basis for the signal subspace these algorithms may be used in a wide range of sensor array applications including bearing estimation beamforming and recursive least squares we present a detailed description of the fst algorithm and its rank adaptive variation ra fst as well as a number of enhancements we also demonstrate the fst s rapid convergence properties in a number of application scenarios

flexible covariance estimation in graphical gaussian models in this paper we propose a class of bayes estimators for the covariance matrix of graphical gaussian models markov with respect to a decomposable graph g working with the w pg family defined by letac and massam ann statist 35 2007 1278 1323 we derive closed form expressions for bayes estimators under the entropy and squared error losses the w pg family includes the classical inverse of the hyper inverse wishart but has many more shape parameters thus allowing for flexibility in differentially shrinking various parts of the covariance matrix moreover using this family avoids recourse to mcmc often infeasible in high dimensional problems we illustrate the performance of our estimators through a collection of numerical examples where we explore frequentist risk properties and the efficacy of graphs in the estimation of high dimensional covariance structures

valid post selection inference it is common practice in statistical data analysis to perform data driven variable selection and derive statistical inference from the resulting model such inference enjoys none of the guarantees that classical statistical theory provides for tests and confidence intervals when the model has been chosen a priori we propose to produce valid post selection inference by reducing the problem to one of simultaneous inference and hence suitably widening conventional confidence and retention intervals simultaneity is required for all linear functions that arise as coefficient estimates in all submodels by purchasing simultaneity insurance for all possible submodels the resulting post selection inference is rendered universally valid under all possible model selection procedures this inference is therefore generally conservative for particular selection procedures but it is always less conservative than full scheffe protection importantly it does not depend on the truth of the selected submodel and hence it produces valid inference even in wrong models we describe the structure of the simultaneous inference problem and give some asymptotic results

projection approximation subspace tracking subspace estimation plays an important role in a variety of modern signal processing applications we present a new approach for tracking the signal subspace recursively it is based on a novel interpretation of the signal subspace as the solution of a projection like unconstrained minimization problem we show that recursive least squares techniques can be applied to solve this problem by making an appropriate projection approximation the resulting algorithms have a computational complexity of o nr where n is the input vector dimension and r is the number of desired eigencomponents simulation results demonstrate that the tracking capability of these algorithms is similar to and in some cases more robust than the computationally expensive batch eigenvalue decomposition relations of the new algorithms to other subspace tracking methods and numerical issues are also discussed

quantization bounds on grassmann manifolds and applications to mimo communications the grassmann manifold gn p l is the set of all p dimensional planes through the origin in the n dimensional euclidean space ln where l is either r or c this paper considers the quantization problem in which a source in gn p l is quantized through a code in gn q l with p and q not necessarily the same the analysis is based on the volume of a metric ball in gn p l with center in gn q l and our chief result is a closed form expression for the volume of a metric ball of radius at most one this volume formula holds for arbitrary n p q and l while previous results pertained only to some special cases based on this volume formula several bounds are derived for the rate distortion tradeoff assuming that the quantization rate is sufficiently high the lower and upper bounds on the distortion rate function are asymptotically identical and therefore precisely quantify the asymptotic rate distortion tradeoff we also show that random codes are asymptotically optimal in the sense that they achieve the minimum possible distortion in probability as n and the code rate approach infinity linearly finally as an application of the derived results to communication theory we quantify the effect of beamforming matrix selection in multiple antenna communication systems with finite rate channel state feedback

computing centre conditions for certain cubic systems abstract we present necessary and sufficient conditions for a critical point of certain two dimensional cubic differential systems to be a centre extensive use of the computer algebra system reduce is involved the search for necessary and sufficient conditions for a centre has long been of considerable interest in the theory of nonlinear differential equations it has proved to be a difficult problem and full conditions are known for very few classes of systems such conditions are also required in the investigation of hilbert s sixteenth problem concerning the number of limit cycles of polynomial systems

poissonian behavior of ising spin systems in an external field we apply the stein chen method for poisson approximation to spin half ising type models in positive external field which satisfy the fkg inequality in particular we show that provided the density of minus spins is low and can be expanded as a convergent power series in the activity fugacity variable the distribution of minus spins is approximately poisson the error of the approximation is inversely proportional to the number of lattice sites we obtain upper and lower bounds on the total variation distance between the exact distribution and its poisson approximation we illustrate these results by application to specific models including the mean field and nearest neighbor ferromagnetic ising models

sphere packing bounds via spherical codes the sphere packing problem asks for the greatest density of a packing of congruent balls in euclidean space the current best upper bound in all suciently high dimensions is due to kabatiansky and levenshtein in 1978 we revisit their argument and improve their bound by a constant factor using a simple geometric argument and we extend the argument to packings in hyperbolic space for which it gives an exponential improvement over the previously known bounds additionally we show that the cohn elkies linear programming bound is always at least as strong as the kabatiansky levenshtein bound this result is analogous to rodemich s theorem in coding theory finally we develop hyperbolic linear programming bounds and prove the analogue of rodemich s theorem there as well

lower bounds on the probability of error for classical and classical quantum channels in this paper lower bounds on error probability in coding for discrete classical and classical quantum channels are studied the contribution of the paper goes in two main directions 1 extending classical bounds of shannon to classical quantum channels and 2 proposing a new framework for lower bounding the probability of error of channels with a zero error capacity in the low rate region the relation between these two problems is revealed by showing that lovasz bound on zero error capacity emerges as a natural consequence of the sphere packing bound once we move to the more general context of classical quantum channels a variation of lovasz bound is then derived to lower bound the probability of error in the low rate region by means of auxiliary channels as a result of this study connections between the lovasz theta function the expurgated bound of gallager the cutoff rate of a classical channel and the sphere packing bound for classical quantum channels are established

poisson approximation for large contours in low temperature ising models we consider the contour representation of the infinite volume ising model at any fixed inverse temperature the solution of 0e 1 let be the infinite volume measure fix v zd 0 and a large n such that calling gn v the set of contours of length at least n intersecting v there are in average contours in gn v under we show that the total variation distance between the law of gn v under and a poisson process is bounded by a constant depending on and times e n the proof builds on the chen stein method as presented by arratia goldstein and gordon the control of the correlations is obtained through the loss network space time representation of contours due to fernandez ferrari and garcia

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

the center conditions for a lienard system we obtain necessary and sufficient coefficient conditions for a center at the origin for a lienard system with nonlinearities of degree six the necessity of the conditions is derived from the first seven focus quantities and their sufficiency is proved by cherkas s method

non parametric detection of the number of signals hypothesis testing and random matrix theory detection of the number of signals embedded in noise is a fundamental problem in signal and array processing this paper focuses on the non parametric setting where no knowledge of the array manifold is assumed first we present a detailed statistical analysis of this problem including an analysis of the signal strength required for detection with high probability and the form of the optimal detection test under certain conditions where such a test exists second combining this analysis with recent results from random matrix theory we present a new algorithm for detection of the number of sources via a sequence of hypothesis tests we theoretically analyze the consistency and detection performance of the proposed algorithm showing its superiority compared to the standard minimum description length mdl based estimator a series of simulations confirm our theoretical analysis

complete characterization of hadamard powers preserving loewner positivity monotonicity and convexity abstract entrywise powers of symmetric matrices preserving positivity monotonicity or convexity with respect to the loewner ordering arise in various applications and have received much attention recently in the literature following fitzgerald and horn 1977 8 it is well known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness similar phenomena have also recently been shown by hiai 2009 to occur for monotonicity and convexity in this paper we complete the characterization of all the entrywise powers below and above the critical exponents that are positive monotone or convex on the cone of positive semidefinite matrices we then extend the original problem by fully classifying the positive monotone or convex powers in a more general setting where additional rank constraints are imposed on the matrices we also classify the entrywise powers that are super sub additive with respect to the loewner ordering finally we extend all the previous characterizations to matrices with negative entries our analysis consequently allows us to answer a question raised by bhatia and elsner 2007 regarding the smallest dimension for which even extensions of the power functions do not preserve loewner positivity

retaining positive definiteness in thresholded matrices abstract positive definite p d matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications in modern high dimensional applications a common approach to finding sparse positive definite matrices is to threshold their small off diagonal elements this thresholding sometimes referred to as hard thresholding sets small elements to zero thresholding has the attractive property that the resulting matrices are sparse and are thus easier to interpret and work with in many applications it is often required and thus implicitly assumed that thresholded matrices retain positive definiteness in this paper we formally investigate the algebraic properties of p d matrices which are thresholded we demonstrate that for positive definiteness to be preserved the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of complete components this result rigorously demonstrates that except in special cases positive definiteness can be easily lost we then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive definiteness when thresholded consequently we derive characterizations of matrices which retain positive definiteness when thresholded with respect to important classes of graphs in particular we demonstrate that retaining positive definiteness upon thresholding is governed by complex algebraic conditions

infinite systems of interacting chains with memory of variable length a stochastic model for biological neural nets we consider a new class of non markovian processes with a countable number of interacting components at each time unit each component can take two values indicating if it has a spike or not at this precise moment the system evolves as follows for each component the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component this class of systems extends in a non trivial way both the interacting particle systems which are markovian spitzer in adv math 5 246 290 1970 and the stochastic chains with memory of variable length which have finite state space rissanen in ieee trans inf theory 29 5 656 664 1983 these features make it suitable to describe the time evolution of biological neural systems we construct a stationary version of the process by using a probabilistic tool which is a kalikow type decomposition either in random environment or in space time this construction implies uniqueness of the stationary process finally we consider the case where the interactions between components are given by a critical directed erdos renyi type random graph with a large but finite number of components in this framework we obtain an explicit upper bound for the correlation between successive inter spike intervals which is compatible with previous empirical findings

wishart distributions for decomposable covariance graph models gaussian covariance graph models encode marginal independence among the components of a multivariate random vector by means of a graph g these models are distinctly different from the traditional concentration graph models often also referred to as gaussian graphical models or covariance selection models since the zeros in the parameter are now reflected in the covariance matrix as compared to the concentration matrix 1 the parameter space of interest for covariance graph models is the cone p g of positive definite matrices with fixed zeros corresponding to the missing edges of g as in letac and massam ann statist 35 2007 1278 1323 we consider the case where g is decomposable in this paper we construct on the cone p g a family of wishart distributions which serve a similar purpose in the covariance graph setting as those constructed by letac and massam ann statist 35 2007 1278 1323 and dawid and lauritzen ann statist 21 1993 1272 1317 do in the concentration graph setting we proceed to undertake a rigorous study of these covariance wishart distributions and derive several deep and useful properties of this class first they form a rich conjugate family of priors with multiple shape parameters for covariance graph models second we show how to sample from these distributions by using a block gibbs sampling algorithm and prove convergence of this block gibbs sampler development of this class of distributions enables bayesian inference which in turn allows for the estimation of e even in the case when the sample size is less than the dimension of the data i e when n p otherwise not generally possible in the maximum likelihood framework third we prove that when g is a homogeneous graph our covariance priors correspond to standard conjugate priors for appropriate directed acyclic graph dag models this correspondence enables closed form expressions for normalizing constants and expected values and also establishes hyper markov properties for our class of priors we also note that when g is homogeneous the family iw qg of letac and massam ann statist 35 2007 1278 1323 is a special case of our covariance wishart distributions fourth and finally we illustrate the use of our family of conjugate priors on real and simulated data

fast and stable yast algorithm for principal and minor subspace tracking this paper presents a new implementation of the yast algorithm for principal and minor subspace tracking yast was initially derived from the subspace projection sp algorithm by davila which was known for its exceptional convergence rate compared with other classical principal subspace trackers the novelty in the yast algorithm was the lower computational cost linear if the data correlation matrix satisfies a so called shift invariance property and the extension to minor subspace tracking however the original implementation of the yast algorithm suffered from a numerical stability problem the subspace weighting matrix slowly loses its orthonormality we thus propose in this paper a new implementation of yast whose stability is established theoretically and tested via numerical simulations this algorithm combines all the desired properties for a subspace tracker remarkably high convergence rate lowest steady state error linear complexity and numerical stability regarding the orthonormality of the subspace weighting matrix

necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix let xk i i 1 k 1 be a double array of nondegenerate i i d random variables and let pn n 1 be a sequence of positive integers such that n pn is bounded away from 0 and in this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry l n rm max 1 leq i 0 we show that the following six statements are equivalent bf i quad lim n to infty n 2 int limits n log n 1 4 infty left f n 1 x f n 1 left frac sqrt n log n x right right df x 0 bf ii quad n mathbb p left max 1 leq i j leq n x 1 i x 1 j geq sqrt n log n right to 0 quad rm as n to infty bf iii quad frac w mu n sqrt n log n stackrel mathbb p rightarrow 2 sigma 2 bf iv quad left frac n log n right 1 2 l n stackrel mathbb p rightarrow 2 bf v quad lim n rightarrow infty mathbb p left frac w mu n 2 n sigma 4 a n leq t right exp left frac 1 sqrt 8 pi e t 2 right infty t infty bf vi quad lim n rightarrow infty mathbb p left n l n 2 a n leq t right exp left frac 1 sqrt 8 pi e t 2 right infty t infty where mu mathbb e x 1 1 sigma 2 mathbb e x 1 1 mu 2 and an 4 log pn log log pn the equivalences between i ii iii and v assume that only mathbb e x 1 1 2 1 2 and n 1 alpha l n stackrel mathbb p rightarrow 0 1 2 alpha leq 1 respectively the current work thus provides weak limit analogues of the strong limit theorems of li and rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of ln obtained by jiang some open problems are also posed

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

spherical cap packing asymptotics and rank extreme detection we study the spherical cap packing problem with a probabilistic approach such probabilistic considerations result in an asymptotic sharp universal uniform bound on the maximal inner product between any set of unit vectors and a stochastically independent uniformly distributed unit vector when the set of unit vectors are themselves independently uniformly distributed we further develop the extreme value distribution limit of the maximal inner product which characterizes its uncertainty around the bound as applications of the above mentioned asymptotic results we derive 1 an asymptotic sharp universal uniform bound on the maximal spurious correlation as well as its uniform convergence in distribution when the explanatory variables are independently gaussian distributed and 2 an asymptotic sharp universal bound on the maximum norm of a low rank elliptically distributed vector as well as related limiting distributions with these results we develop a fast detection method for a low rank structure in high dimensional gaussian data without using the spectrum information

coding for errors and erasures in random network coding the problem of error control in random linear network coding is considered a ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic motivated by the property that linear network coding is vector space preserving information transmission is modeled as the injection into the network of a basis for a vector space v and the collection by the receiver of a basis for a vector space u a metric on the projective geometry associated with the packet space is introduced and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space v capu is sufficiently large if the dimension of each codeword is restricted to a fixed integer the code forms a subset of a finite field grassmannian or equivalently a subset of the vertices of the corresponding grassmann graph sphere packing and sphere covering bounds as well as a generalization of the singleton bound are provided for such codes finally a reed solomon like code construction related to gabidulin s construction of maximum rank distance codes is described and a sudan style ldquolist 1rdquo minimum distance decoding algorithm is provided

on the integrability of two dimensional flows this paper deals with the notion of integrability of flows or vector fields on two dimensional manifolds we consider the following two key points about first integrals 1 they must be defined on the whole domain of definition of the flow or vector field or defined on the complement of some special orbits of the system 2 how are they computed we prove that every local flow on a two dimensional manifold m always has a continuous first integral on each component of m where is the set of all separatrices of we consider the inverse integrating factor and we show that it is better to work with it instead of working directly with a first integral or an integrating factor for studying the integrability of a given two dimensional flow or vector field finally we prove the existence and uniqueness of an analytic inverse integrating factor in a neighborhood of a strong focus of a non resonant hyperbolic node and of a siegel hyperbolic saddle

nondegenerate centers for abel polynomial differential equations of second kind in this paper we study the center problem for abel polynomial differential equations of second kind computing the focal values and using modular arithmetics and grbner bases we find the center conditions for such systems for lower degrees

bounds on packings of spheres in the grassmann manifold we derive the gilbert varshamov and hamming bounds for packings of spheres codes in the grassmann manifolds over r and c asymptotic expressions are obtained for the geodesic metric and projection frobenius chordal metric on the manifold

two algorithms for fast approximate subspace tracking new fast algorithms are presented for tracking singular values singular vectors and the dimension of a signal subspace through an overlapping sequence of data matrices the basic algorithm is called fast approximate subspace tracking fast the algorithm is derived for the special case in which the matrix is changed by deleting the oldest column shifting the remaining columns to the left and adding a new column on the right a second algorithm fast2 is specified by modifying fast to trade reduced accuracy for higher speed the speed and accuracy are compared with the pl algorithm the past and pastd algorithms and the fst algorithm an extension to multicolumn updates for the fast algorithm is also discussed

loss network representation of peierls contours we present a probabilistic approach for the study of systems with exclusions in the regime traditionally studied via cluster expansion methods in this paper we focus on its application for the gases of peierls contours found in the study of the ising model at low temperatures but most of the results are general we realize the equilibrium measure as the invariant measure of a loss network process whose existence is ensured by a subcriticality condition of a dominant branching process in this regime the approach yields besides existence and uniqueness of the measure properties such as exponential space convergence and mixing and a central limit theorem the loss network converges exponentially fast to the equilibrium measure without metastable traps this convergence is faster at low temperatures where it leads to the proof of an asymptotic poisson distribution of contours our results on the mixing properties of the measure are comparable to those obtained with duplicated variables expansion used to treat systems with disorder and coupled map lattices it works in a larger region of validity than usual cluster expansion formalisms and it is not tied to the analyticity of the pressure in fact it does not lead to any kind of expansion for the latter and the properties of the equilibrium measure are obtained without resorting to combinatorial or complexanalysis techniques

dobrushin kotecký shlosman theorem for polygonal markov fields in the plane we consider the so called length interacting arak surgailis polygonal markov fields with v shaped nodes a continuum and isometry invariant process in the plane sharing a number of properties with the two dimensional ising model for these polygonal fields we establish a low temperature phase separation theorem in the spirit of the dobrushin koteck shlosman theory with the corresponding wulff shape deteremined to be a disk due to the rotation invariant nature of the considered model as an important tool replacing the classical cluster expansion techniques and very well suited for our geometric setting we use a graphical construction built on contour birth and death process following the ideas of fernandez ferrari and garcia

asymptotic distribution of the largest off diagonal entry of correlation matrices suppose that we have observations from a dimensional population we are interested in testing that the variates of the population are independent under the situation where goes to infinity as a test statistic is chosen to be where is the sample correlation coefficient between the th coordinate and the th coordinate of the population under an independent hypothesis we prove that the asymptotic distribution of is an extreme distribution of type by using the chen stein poisson approximation method and the moderate deviations for sample correlation coefficients as a statistically more relevant result a limit distribution for where is spearman s rank correlation coefficient between the th coordinate and the th coordinate of the population is derived

an approach to solving systems of polynomials via modular arithmetics with applications the objective of this paper is twofold first we describe a method to solve large systems of polynomial equations using modular arithmetics then we apply the approach to the study of the problem of linearizability for a quadratic system of ordinary differential equations

random walk kernels and learning curves for gaussian process regression on random graphs we consider learning on graphs guided by kernels that encode similarity between vertices our focus is on random walk kernels the analogues of squared exponential kernels in euclidean spaces we show that on large locally treelike graphs these have some counter intuitive properties specifically in the limit of large kernel lengthscales we consider using these kernels as covariance functions of gaussian processes in this situation one typically scales the prior globally to normalise the average of the prior variance across vertices we demonstrate that in contrast to the euclidean case this generically leads to significant variation in the prior variance across vertices which is undesirable from a probabilistic modelling point of view we suggest the random walk kernel should be normalised locally so that each vertex has the same prior variance and analyse the consequences of this by studying learning curves for gaussian process regression numerical calculations as well as novel theoretical predictions for the learning curves using belief propagation show that one obtains distinctly different probabilistic models depending on the choice of normalisation our method for predicting the learning curves using belief propagation is significantly more accurate than previous approximations and should become exact in the limit of large random graphs

wishart distributions for decomposable graphs when considering a graphical gaussian model n g markov with respect to a decomposable graph g the parameter space of interest for the precision parameter is the cone p g of positive definite matrices with fixed zeros corresponding to the missing edges of g the parameter space for the scale parameter of n g is the cone q g dual to p g of incomplete matrices with submatrices corresponding to the cliques of g being positive definite in this paper we construct on the cones q g and p g two families of wishart distributions namely the type i and type ii wisharts they can be viewed as generalizations of the hyper wishart and the inverse of the hyper inverse wishart as defined by dawid and lauritzen ann statist 21 1993 1272 1317 we show that the type i and ii wisharts have properties similar to those of the hyper and hyper inverse wishart indeed the inverse of the type ii wishart forms a conjugate family of priors for the covariance parameter of the graphical gaussian model and is strong directed hyper markov for every direction given to the graph by a perfect order of its cliques while the type i wishart is weak hyper markov moreover the inverse type ii wishart as a conjugate family presents the advantage of having a multidimensional shape parameter thus offering flexibility for the choice of a prior both type i and ii wishart distributions depend on multivariate shape parameters a shape parameter is acceptable if and only if it satisfies a certain eigenvalue property we show that the sets of acceptable shape parameters for a noncomplete g have dimension equal to at least one plus the number of cliques in g these families as conjugate families are richer than the traditional diaconis ylvisaker conjugate families which all have a shape parameter set of dimension one a decomposable graph which does not contain a three link chain as an induced subgraph is said to be homogeneous in this case our wisharts are particular cases of the wisharts on homogeneous cones as defined by andersson and wojnar j theoret probab 17 2004 781 818 and the dimension of the shape parameter set is even larger than in the nonhomogeneous case it is indeed equal to the number of cliques plus the number of distinct minimal separators using the model where g is a three link chain we show by computing a 7 tuple integral that in general we cannot expect the shape parameter sets to have dimension larger than the number of cliques plus one

infinite dimensional srb measures we review the basic steps leading to the construction of a sinai ruelle bowen srb measure for an infinite lattice of weakly coupled expanding circle maps and we show that this measure has exponential decay of space time correlations first using the perron frobenius operator one connects the dynamical system of coupled maps on a d dimensional lattice to an equilibrium statistical mechanical model on a lattice of dimension d 1 this lattice model is for weakly coupled maps in a high temperature phase and we use a general but very elementary method to prove exponential decay of correlations at high temperatures

optimal estimation and rank detection for sparse spiked covariance matrices this paper considers a sparse spiked covariance matrix model in the high dimensional setting and studies the minimax estimation of the covariance matrix and the principal subspace as well as the minimax rank detection the optimal rate of convergence for estimating the spiked covariance matrix under the spectral norm is established which requires significantly different techniques from those for estimating other structured covariance matrices such as bandable or sparse covariance matrices we also establish the minimax rate under the spectral norm for estimating the principal subspace the primary object of interest in principal component analysis in addition the optimal rate for the rank detection boundary is obtained this result also resolves the gap in a recent paper by berthet and rigollet ann stat 41 4 1780 1815 2013 where the special case of rank one is considered

geometric representation of high dimension low sample size data summary high dimension low sample size data are emerging in various areas of science we find a common structure underlying many such data sets by using a non standard type of asymptotics the dimension tends to while the sample size is fixed our analysis shows a tendency for the data to lie deterministically at the vertices of a regular simplex essentially all the randomness in the data appears only as a random rotation of this simplex this geometric representation is used to obtain several new statistical insights

large scale correlation screening this article addresses the problem of screening for variables with high correlations in high dimensional data in which there can be many fewer samples than variables we focus on threshold based correlation screening methods for three related applications screening for variables with large correlations within a single treatment autocorrelation screening screening for variables with large cross correlations over two treatments cross correlation screening and screening for variables that have persistently large autocorrelations over two treatments persistent correlation screening the novelty of correlation screening is that it identifies a smaller number of variables that are highly correlated with others compared with identifying a number of correlation parameters correlation screening suffers from a phase transition phenomenon as the correlation threshold decreases the number of discoveries increases abruptly we obtain asymptotic expressions for the mean number of discoveries and the phase t

taming griffiths singularities infinite differentiability of quenched correlation functions we prove infinite differentiability of the magnetization and of all quenched correlation functions for disordered spin systems at high temperature or strong magnetic field in the presence of griffiths singularities we also show uniqueness of the gibbs state and exponential decay of truncated correlation functions with probability one our results are obtained through new simple modified high temperature or low activity expansions whose convergence can be displayed by elementary probabilistic arguments our results require no assumptions on the probability distributions of the random parameters except for the obvious one of no percolation of infinite couplings and in the strong field situation for the also obvious requirement that zero magnetic fields do not percolate

improved upper bounds for the contact process critical value the best known upper bound for the critical value c of the basic one dimensional contact process is 2 most techniques for finding bounds on critical values have the property that they can be modified in order to obtain improved bounds this seemed not to be the case for the approach which yielded c 2 for the basic contact process in this paper we propose a technique for generating better bounds in this context to illustrate its use we carry out the full program in one case with the conclusion that c 1 942

a unified approach to phase diagrams in field theory and statistical mechanics we construct the phase diagram of any system which admits a low temperature polymer or cluster expansion such an expansion turns the system into a hard core interacting contour model with small but not necessarily positive activities the method uses some of zahradnik s ideas z1 but applies equally well to systems with complex interactions we give two applications first to low temperaturep 2 models with complex couplings and second to a computation of asymptotics of partition functions in periodic volumes if the index of a supersymmetric field theory is known the second application would help determine the number of phases in infinite volume

sphere packing bounds in the grassmann and stiefel manifolds applying the riemann geometric machinery of volume estimates in terms of curvature bounds for the minimal distance of packings codes in the grassmann and stiefel manifolds will be derived and analyzed in the context of space time block codes this leads to a monotonically increasing minimal distance lower bound as a function of the block length this advocates large block lengths for the code design

beamforming codebooks for two transmit antenna systems based on optimum grassmannian packings precoding codebook design for limited feedback mimo systems is known to reduce to a discretization problem on a grassmann manifold the case of two antenna beamforming is special in that it is equivalent to quantizing the real sphere the isometry between the grassmannian g2 1 and the real sphere s2 shows that discretization problems in the grassmannian g2 1 are directly solved by corresponding spherical codes notably the grassmannian line packing problem in 2 namely maximizing the minimum distance is equivalent to the tammes problem on the real sphere so that optimum spherical packings give optimum grassmannian packings moreover a simple isomorphism between g2 1 and s2 enables to analytically derive simple codebooks in closed form having low implementation complexity using the simple geometry of some of these codebooks we derive closed form expressions of the probability density function of the relative snr loss due to limited feedback we also investigate codebooks based on other spherical arrangements such as solutions maximizing the harmonic mean of the mutual distances among the codewords which is known as the thomson problem we find that in some special cases grassmannian codebooks based on these other spherical arrangements outperform codebooks from grassmannian packing

cluster and virial expansions for the multi species tonks gas we consider a mixture of non overlapping rods of different lengths ell k moving in mathbb r or mathbb z our main result are necessary and sufficient convergence criteria for the expansion of the pressure in terms of the activities z k and the densities rho k this provides an explicit example against which to test known cluster expansion criteria and illustrates that for non negative interactions the virial expansion can converge in a domain much larger than the activity expansion in addition we give explicit formulas that generalize the well known relation between non overlapping rods and labelled rooted trees we also prove that for certain choices of the activities the system can undergo a condensation transition akin to that of the zero range process the key tool is a fixed point equation for the pressure

invariance under quasi isometries of subcritical and supercritical behavior in the boolean model of percolation in this work we study the poisson boolean model of percolation in locally compact polish metric spaces and we prove the invariance of subcritical and supercritical phases under mm quasi isometries more precisely we prove that if a metric space m is mm quasi isometric to another metric space n and the poisson boolean model in m exhibits any of the following a a subcritical phase b a supercritical phase or c a phase transition then respectively so does the poisson boolean model of percolation in n then we use these results in order to understand the phase transition phenomenon in a large family of metric spaces indeed we study the poisson boolean model of percolation in the context of riemannian manifolds in a large family of nilpotent lie groups and in cayley graphs also we prove the existence of a subcritical phase in gromov spaces with bounded growth at some scale

kukles revisited advances in computing techniques we revisit the kukles system to show how advances in computer hardware and software have improved symbolic calculations we confirm directly that there are no non persistent centres for the kukles system we also prove an earlier conjecture regarding the isochronous centres for the extended kukles system we introduce a technique whereby modular resultants can be used to investigate the properties of resultants that cannot be readily calculated using the existing software

on the central limit theorem for stationary mixing random fields a simple proof of a central limit theorem for stationary random fields under mixing conditions is given generalizing some results obtained by more complicated methods e g bernstein s method

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

on beamforming with finite rate feedback in multiple antenna systems we study a multiple antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations assuming that the transmitter uses the quantized information for beamforming we derive a universal lower bound on the outage probability for any finite set of beamformers the universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel using the bound it is shown that finite information systems approach the perfect information case as t 1 2 sup b t 1 where b is the number of feedback bits and t is the number of transmit antennas the geometrical bounding technique used in the proof of the lower bound also leads to a design criterion for good beamformers whose outage performance approaches the lower bound the design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook and is equivalent to the problem of designing unitary space time codes under certain conditions finally we show that good beamformers are good packings of two dimensional subspaces in a 2t dimensional real grassmannian manifold with chordal distance as the metric

centers of planar generalized abel equations abstract we deal with the differential equation r d r d a r n b r m where r are the polar coordinates in the plane r 2 m and n are integers such that m n 2 and a b are c 1 functions note that when n 2 and m 3 we have an abel differential equation for this class of generalized abel equations we characterize a new family of centers