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

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

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

random generation of stochastic area integrals the authors describe a method of random generation of the integrals a 1 2 t t h int t t h int t s dw 1 r dw 2 s int t t h int t s dw 2 r dw 1 s together with the increments w 1 t h w 1 t and w 2 t h w 2 t of a two dimensional brownian path w 1 t w 2 t the method chosen is based on marsaglia s rectangle wedge tail method generalised to higher dimensions the motivation is the need for a numerical scheme for simulation of strong solutions of general multidimensional stochastic differential equations with an order of convergence o h where h is the stepsize previously no method has obtained an order of convergence better than o sqrt h in the general case

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

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

variable step size control in the numerical solution of stochastic differential equations we introduce a variable step size method for the numerical approximation of pathwise solutions to stochastic differential equations sdes the method which is dependent on a representation of bro

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

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

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

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

stochastic bifurcations in the nonlinear parallel ising model we investigate the phase transitions of a nonlinear parallel version of the ising model characterized by an antiferromagnetic linear coupling and ferromagnetic nonlinear one this model arises in problems of opinion formation the mean field approximation shows chaotic oscillations by changing the couplings or the connectivity the spatial model shows bifurcations in the average magnetization similar to that seen in the mean field approximation induced by the change of the topology after rewiring short range to long range connection as predicted by the small world effect these coherent periodic and chaotic oscillations of the magnetization reflect a certain degree of synchronization of the spins induced by long range couplings similar bifurcations may be induced in the randomly connected model by changing the couplings or the connectivity and also the dilution degree of asynchronism of the updating we also examined the effects of inhomogeneity mixing ferromagnetic and antiferromagnetic coupling which induces an unexpected bifurcation diagram with a bubbling behavior as also happens for dilution

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

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

the noisy voter model under the influence of contrarians abstract the influence of contrarians on the noisy voter model is studied at the mean field level the noisy voter model is a variant of the voter model where agents can adopt two opinions optimistic or pessimistic and can change them by means of an imitation herding and an intrinsic noise mechanisms an ensemble of noisy voters undergoes a finite size phase transition upon increasing the relative importance of the noise to the herding form a bimodal phase where most of the agents share the same opinion to a unimodal phase where almost the same fraction of agent are in opposite states by the inclusion of contrarians we allow for some voters to adopt the opposite opinion of other agents anti herding we first consider the case of only contrarians and show that the only possible steady state is the unimodal one more generally when voters and contrarians are present we show that the bimodal unimodal transition of the noisy voter model prevails only if the number of contrarians in the system is smaller than four and their characteristic rates are small enough for the number of contrarians bigger or equal to four the voters and the contrarians can be seen only in the unimodal phase moreover if the number of voters and contrarians as well as the noise and herding rates are of the same order then the probability functions of the steady state are very well approximated by the gaussian distribution

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

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

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

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

bifurcations in models of a society of reasonable contrarians and conformists we study models of a society composed of a mixture of conformist and reasonable contrarian agents that at any instant hold one of two opinions conformists tend to agree with the average opinion of their neighbors and reasonable contrarians tend to disagree but revert to a conformist behavior in the presence of an overwhelming majority in line with psychological experiments the model is studied in the mean field approximation and on small world and scale free networks in the mean field approximation a large fraction of conformists triggers a polarization of the opinions a pitchfork bifurcation while a majority of reasonable contrarians leads to coherent oscillations with an alternation of period doubling and pitchfork bifurcations up to chaos similar scenarios are obtained by changing the fraction of long range rewiring and the parameter of scale free networks related to the average connectivity

higher order time integration of coulomb collisions in a plasma using langevin equations the extension of langevin equation monte carlo algorithms for coulomb collisions from the conventional euler maruyama time integration to the next higher order of accuracy the milstein scheme has been developed implemented and tested this extension proceeds via a formulation of the angular scattering directly as stochastic differential equations in the fixed frame spherical coordinate velocity variables results from the numerical implementation show the expected improvement o dt vs o dt 1 2 in the strong convergence rate both for the speed v and angular components of the scattering an important result is that this improved convergence is achieved for the angular component of the scattering if and only if the area integral terms in the milstein scheme are included the resulting milstein scheme is of value as a step towards algorithms with both improved accuracy and efficiency these include both algorithms with improved convergence in the averages weak convergence and multi time level schemes the latter have been shown to give a greatly reduced cost for a given overall error level when compared with conventional monte carlo schemes and their performance is improved considerably when the milstein algorithm is used for the underlying time advance versus the euler maruyama algorithm a new method for sampling the area integrals is given which is a simplification of an earlier direct method and which retains high accuracy this method while being useful in its own right because of its relative simplicity is also expected to considerably reduce the computational requirements for the direct conditional sampling of the area integrals that is needed for adaptive strong integration

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

clusters and the entropy in opinion dynamics on complex networks abstract in this work we investigate a heterogeneous population in the modified hegselmann krause opinion model on complex networks we introduce the shannon information entropy about all relative opinion clusters to characterize the cluster profile in the final configuration independent of network structures there exists the optimal stubbornness of one subpopulation for the largest number of clusters and the highest entropy besides there is the optimal bounded confidence or subpopulation ratio of one subpopulation for the smallest number of clusters and the lowest entropy however network structures affect cluster profiles indeed a large average degree favors consensus for making different networks more similar with complete graphs the network size has limited impact on cluster profiles of heterogeneous populations on scale free networks but has significant effects upon those on small world networks

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

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

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