influential features pca for high dimensional clustering we consider a clustering problem where we observe feature vectors x i in r p i 1 2 ldots n from k possible classes the class labels are unknown and the main interest is to estimate them we are primarily interested in the modern regime of p gg n where classical clustering methods face challenges we propose influential features pca if pca as a new clustering procedure in if pca we select a small fraction of features with the largest kolmogorov smirnov ks scores obtain the first k 1 left singular vectors of the post selection normalized data matrix and then estimate the labels by applying the classical k means procedure to these singular vectors in this procedure the only tuning parameter is the threshold in the feature selection step we set the threshold in a data driven fashion by adapting the recent notion of higher criticism as a result if pca is a tuning free clustering method we apply if pca to 10 gene microarray data sets the method has competitive performance in clustering especially in three of the data sets the error rates of if pca are only 29 or less of the error rates by other methods we have also rediscovered a phenomenon on empirical null by efron j amer statist assoc 99 2004 96 104 on microarray data with delicate analysis especially post selection eigen analysis we derive tight probability bounds on the kolmogorov smirnov statistics and show that if pca yields clustering consistency in a broad context the clustering problem is connected to the problems of sparse pca and low rank matrix recovery but it is different in important ways we reveal an interesting phase transition phenomenon associated with these problems and identify the range of interest for each

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

phase transitions for high dimensional clustering and related problems consider a two class clustering problem where we observe x i ell i mu z i z i stackrel mathit i i d sim n 0 i p 1 leq i leq n the feature vector mu in r p is unknown but is presumably sparse the class labels ell i in 1 1 are also unknown and the main interest is to estimate them we are interested in the statistical limits in the two dimensional phase space calibrating the rarity and strengths of useful features we find the precise demarcation for the region of impossibility and region of possibility in the former useful features are too rare weak for successful clustering in the latter useful features are strong enough to allow successful clustering the results are extended to the case of colored noise using le cam s idea on comparison of experiments we also extend the study on statistical limits for clustering to that for signal recovery and that for global testing we compare the statistical limits for three problems and expose some interesting insight we propose classical pca and important features pca if pca for clustering for a threshold t 0 if pca clusters by applying classical pca to all columns of x with an l 2 norm larger than t we also propose two aggregation methods for any parameter in the region of possibility some of these methods yield successful clustering we discover a phase transition for if pca for any threshold t 0 let xi t be the first left singular vector of the post selection data matrix the phase space partitions into two different regions in one region there is a t such that cos xi t ell rightarrow 1 and if pca yields successful clustering in the other cos xi t ell leq c 0 0 our results require delicate analysis especially on post selection random matrix theory and on lower bound arguments

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

efficient isoparametric trimmed hexahedral elements with explicit shape functions abstract in this study we develop isoparametric trimmed hexahedral th elements with explicit shape functions for finite element analysis using trimmed meshes th elements are placed at the boundary of a trimmed mesh generated by cutting a regular hexahedral mesh with a geometric surface master th elements are defined based on the marching cubes algorithm for an isoparametric mapping from the reference coordinates to the physical coordinates shape function values at the support points of master th elements are precomputed by solving laplace s equation with appropriate boundary conditions explicit shape functions for master th elements are constructed by interpolating the precomputed shape function values at the support points of master th elements the present explicit shape functions satisfy the required properties of fem and significantly reduce the computational cost compared with the previously developed polyhedral shape functions

asymptotics of empirical eigenstructure for high dimensional spiked covariance we derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime which takes into account the magnitude of spiked eigenvalues sample size and dimensionality this regime allows high dimensionality and diverging eigenvalues and provides new insights into the roles that the leading eigenvalues sample size and dimensionality play in principal component analysis our results are a natural extension of those in paul 2007 to a more general setting and solve the rates of convergence problems in shen et al 2013 they also reveal the biases of estimating leading eigenvalues and eigenvectors by using principal component analysis and lead to a new covariance estimator for the approximate factor model called shrinkage principal orthogonal complement thresholding s poet that corrects the biases our results are successfully applied to outstanding problems in estimation of risks of large portfolios and false discovery proportions for dependent test statistics and are illustrated by simulation studies

breather solutions for inhomogeneous fpu models using birkhoff normal forms abstract we present results on spatially localized oscillations in some inhomogeneous nonlinear lattices of fermi pasta ulam fpu type derived from phenomenological nonlinear elastic network models proposed to study localized protein vibrations the main feature of the fpu lattices we consider is that the number of interacting neighbors varies from site to site and we see numerically that this spatial inhomogeneity leads to spatially localized normal modes in the linearized problem this property is seen in 1 d models and in a 3 d model with a geometry obtained from protein data the spectral analysis of these examples suggests some non resonance assumptions that we use to show the existence of invariant subspaces of spatially localized solutions in quartic birkhoff normal forms of the fpu systems the invariant subspaces have an additional symmetry and this fact allows us to compute periodic orbits of the quartic normal form in a relatively simple way

an evaluation of mixed order versus full order vector finite elements recently proposed vector finite elements are compared to previously published elements and independently implemented and validated in the context of rectangular waveguide analysis both mixed order and full order elements up to order two are considered the use of quadrature cubature to evaluate the finite element system matrices is discussed details of some analytical work required before quadrature is applied are presented for several types of basis functions for many rf applications mixed order elements offer optimal accuracy for a given number of degrees of freedom and an empty through waveguide solution verifies this with similar convergence rates for mixed and full order elements of the same order however a capacitive iris in the rectangular waveguide with a solution dominated by the quasi static electric field motivates the use of full order elements in such cases results for the electric fields in the vicinity of the iris also demonstrate the enhanced accuracy of the full order elements here this is further illustrated using a dielectric load in a waveguide where complete order elements produce a better approximation of the normal jump discontinuity although this paper addresses guided wave problems the results are also applicable to radiation and scattering problems

instabilities of breathers in a finite nls lattice abstract we study some aspects of the dynamics of unstable breathers in a three site discrete cubic nls chain with dirichlet boundary conditions we view breathers as fixed points of the energy in the reduced phase space obtained by eliminating directions related to the global phase symmetry of the system we use a combination of numerical calculations and morse theoretical arguments to see that there are two breathers that correspond to critical energies where the energy hypersurface changes its connectivity these breathers are elliptic hyperbolic fixed points of the reduced four dimensional system we compute the periodic orbits in their center manifolds lyapunov orbits and see evidence for homoclinic intersections of their stable and unstable manifolds we also examine the possibility of heteroclinic connections between lyapunov orbits these however appear not to exist for the energies near the energy where the energy hypersurface becomes connected

fifteen node tetrahedral elements for explicit methods in nonlinear solid dynamics abstract despite the ease in meshing and benefits for modeling flexure curved shapes etc second order tetrahedral elements are not contained in typical explicit solid dynamic programs this is primarily due to the lack of both a satisfactory consistent nodal loading distribution and mass lumping technique row summation lumping for instance produces negative vertex node masses for the popular ten node serendipity tetrahedron which also has zero vertex node loads resulting from a constant pressure on an element face this has led to piecewise composites of four node tetrahedrons to represent a ten node one in explicit codes in this paper truly second order fifteen node formulations for compressible and for nearly incompressible materials are presented and evaluated in addition to producing all positive nodal loads from a uniform traction row summation mass lumping for the fifteen node element is shown to produce all positive nodal masses performance is assessed in standard benchmark problems and practical applications using various elastic and elastic plastic material models and involving very large strains deformations severe distortions and contact impact comparisons are also made with several first order elements and second order hexahedral formulations the offered elements performed satisfactorily in all examples as recently found for second order hexahedral elements it is shown that the inclusion of face and centroidal nodes is vital for robust performance with row summation lumping and high order quadrature rules are crucial with explicit methods these second order elements are shown to be viable for practical applications especially using today s parallel computers whereas the reliable performance is generally attained at significant computational expense compared with first order and brick types these elements can be more computationally competitive in flexure and have the desirable trait that they are amenable to automatic tetrahedral meshing software

pca consistency in high dimension low sample size context principal component analysis pca is an important tool of dimension reduction especially when the dimension or the number of variables is very high asymptotic studies where the sample size is fixed and the dimension grows i e high dimension low sample size hdlss are becoming increasingly relevant we investigate the asymptotic behavior of the principal component pc directions hdlss asymptotics are used to study consistency strong inconsistency and subspace consistency we show that if the first few eigenvalues of a population covariance matrix are large enough compared to the others then the corresponding estimated pc directions are consistent or converge to the appropriate subspace subspace consistency and most other pc directions are strongly inconsistent broad sets of sucient conditions for each of these cases are specified and the main theorem gives a catalogue of possible combinations in preparation for these results we show that the geometric representation of hdlss data holds under general conditions which includes a mixing condition and a broad range of sphericity measures of the covariance matrix

optimal rates of convergence for sparse covariance matrix estimation this paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and bregman divergence losses a major focus is on the derivation of a rate sharp minimax lower bound the problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems standard techniques fail to yield good results and new tools are thus needed we first develop a lower bound technique that is particularly well suited for treating two directional problems such as estimating sparse covariance matrices the result can be viewed as a generalization of le cam s method in one direction and assouad s lemma in another this lower bound technique is of independent interest and can be used for other matrix estimation problems we then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique a thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm the results are then extended to

high dimensional covariance matrix estimation using a factor model abstract high dimensionality comparable to sample size is common in many statistical problems we examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases motivated by the arbitrage pricing theory in finance a multi factor model is employed to reduce dimensionality and to estimate the covariance matrix the factors are observable and the number of factors k is allowed to grow with p we investigate the impact of p and k on the performance of the model based covariance matrix estimator under mild assumptions we have established convergence rates and asymptotic normality of the model based estimator its performance is compared with that of the sample covariance matrix we identify situations under which the factor approach increases performance substantially or marginally the impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied the asymptotic results are supported by a thorough simulation study

integration by table look up for p version finite elements on curved tetrahedra abstract we develop a method of evaluating element level matrices and vectors associated with p version finite element computations on tetrahedral meshes with curved boundaries the procedure uses optimal interpolation formula q chen i babuska approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle comput methods appl mech engrg 128 1995 405 417 q chen i babuska approximate optimal polynomial interpolation points in the tetrahedron comput methods appl mech engrg 137 1997 89 94 for non polynomial portions of integrands followed by exact integration exact integrals of product combinations of polynomial interpolants and shape functions are stored so that different integrals may be efficiently evaluated by a table look up we present error analysis for our scheme applied to curvilinear tetrahedral meshes with exact geometric representation of the boundary of the domain as defined in a geometric modelling system numerical results are presented for problems involving the helmholtz equation

estimating sparse precision matrix optimal rates of convergence and adaptive estimation precision matrix is of significant importance in a wide range of applications in multivariate analysis this paper considers adaptive minimax estimation of sparse precision matrices in the high dimensional setting optimal rates of convergence are established for a range of matrix norm losses a fully data driven estimator based on adaptive constrained l1 minimization is proposed and its rate of convergence is obtained over a collection of parameter spaces the estimator called aclime is easy to implement and performs well numerically a major step in establishing the minimax rate of convergence is the derivation of a rate sharp lower bound a two directional lower bound technique is applied to obtain the minimax lower bound the upper and lower bounds together yield the optimal rates of convergence for sparse precision matrix estimation and show that the aclime estimator is adaptively minimax rate optimal for a collection of parameter spaces and a range of matrix norm losses simultaneously

a straightforward structure to construct shape functions for variable p order meshes this paper describes a method for specifying variable p order meshes based on the topological hierarchy of mesh entities defining the finite elements a decomposition of p version mesh entity shape functions into a product of a function defined over the mesh entity and a blending function defined over the elements that the mesh entity bounds is presented the decomposed shape functions allow the effective construction of the shape functions and derivatives needed to construct element matrices for variable p enrichment

exact integrations of polynomials and symmetric quadrature formulas over arbitrary polyhedral grids this paper is concerned with two important elements in the high order accurate spatial discretization of finite volume equations over arbitrary grids one element is the integration of basis functions over arbitrary domains which is used in expressing various spatial integrals in terms of discrete unknowns the other consists of quadrature approximations to those integrals only polynomial basis functions applied to polyhedral and polygonal grids are treated here nontriangular polygonal faces are subdivided into a union of planar triangular facets and the resulting triangulated polyhedron is subdivided into a union of tetrahedra the straight line segment triangle and tetrahedron are thus the fundamental shapes that are the building blocks for all integrations and quadrature approximations integrals of products up to the fifth order are derived in a unified manner for the three fundamental shapes in terms of the position vectors of vertices results are given both in terms of tensor products and products of cartesian coordinates the exact polynomial integrals are used to obtain symmetric quadrature approximations of any degree of precision up to five for arbitrary integrals over the three fundamental domains using a coordinate free formulation simple and rational procedures are developed to derive virtually all quadrature formulas including some previously unpublished four symmetry groups of quadrature points are introduced to derive gauss formulas while their limiting forms are used to derive lobatto formulas representative gauss and lobatto formulas are tabulated the relative efficiency of their application to polyhedral and polygonal grids is detailed the extension to higher degrees of precision is discussed

random perturbation of low rank matrices improving classical bounds abstract matrix perturbation inequalities such as weyl s theorem concerning the singular values and the davis kahan theorem concerning the singular vectors play essential roles in quantitative science in particular these bounds have found application in data analysis as well as related areas of engineering and computer science in many situations the perturbation is assumed to be random and the original matrix has certain structural properties such as having low rank we show that in this scenario classical perturbation results such as weyl and davis kahan can be improved significantly we believe many of our new bounds are close to optimal and also discuss some applications

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

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

numerical modeling of the mixing flow of second order fluids with helical ribbon impellers rheology is known to have a strong impact on the flow behavior and the power consumption of mechanically agitated tanks in the case of viscoelastic fluids the role of elasticity on power draw has yet to be elucidated though recent studies with helical ribbon impellers indicate that elasticity increases torque the objective of this paper is to show that in the case of second order fluids the use of a simple constitutive equation derived from a second order retarded motion expansion succeeds in predicting a rise in power draw owing to elasticity the equations of change governing fluid flow are solved using a finite element method combined with an augmented lagrangian method for the treatment of the non linear constitutive equation we show in particular how the underlying non linear tensor equations can be solved directly using a spectral decomposition of the related matrix operator first the numerical methodology is explained next simulation results are presented in the case of a tank provided with a helical ribbon these results are then compared with experimental data a rise of power draw similar in shape but smaller than that measured experimentally may be noted when elasticity is increased

robust covariance estimation for approximate factor models in this paper we study robust covariance estimation under the approximate factor model with observed factors we propose a novel framework to first estimate the initial joint covariance matrix of the observed data and the factors and then use it to recover the covariance matrix of the observed data we prove that once the initial matrix estimator is good enough to maintain the element wise optimal rate the whole procedure will generate an estimated covariance with desired properties for data with only bounded fourth moment we propose to use adaptive huber loss minimization to give the initial joint covariance estimation this approach is applicable to a much wider class of distributions beyond sub gaussian and elliptical distributions we also present an asymptotic result for adaptive huber s m estimator with a diverging parameter the conclusions are demonstrated by extensive simulations and real data analysis

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

sparse pca optimal rates and adaptive estimation principal component analysis pca is one of the most commonly used statistical procedures with a wide range of applications this paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting under mild technical conditions we first establish the optimal rates of convergence for estimating the principal subspace which are sharp with respect to all the parameters thus providing a complete characterization of the difficulty of the estimation problem in term of the convergence rate the lower bound is obtained by calculating the local metric entropy and an application of fano s lemma the rate optimal estimator is constructed using aggregation which however might not be computationally feasible we then introduce an adaptive procedure for estimating the principal subspace which is fully data driven and can be computed efficiently it is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces a key idea in our construction is a reduction scheme which reduces the sparse pca problem to a high dimensional multivariate regression problem this method is potentially also useful for other related problems

computational and statistical boundaries for submatrix localization in a large noisy matrix we study in this paper computational and statistical boundaries for submatrix localization given one observation of one or multiple nonoverlapping signal submatrix of magnitude lambda and size k m times k n embedded in a large noise matrix of size m times n the goal is to optimal identify the support of the signal submatrix computationally and statistically two transition thresholds for the signal to noise ratio lambda sigma are established in terms of m n k m and k n the first threshold sf snr c corresponds to the computational boundary we introduce a new linear time spectral algorithm that identifies the submatrix with high probability when the signal strength is above the threshold sf snr c below this threshold it is shown that no polynomial time algorithm can succeed in identifying the submatrix under the hidden clique hypothesis the second threshold sf snr s captures the statistical boundary below which no method can succeed in localization with probability going to one in the minimax sense the exhaustive search method successfully finds the submatrix above this threshold in marked contrast to submatrix detection and sparse pca the results show an interesting phenomenon that sf snr c is always significantly larger than sf snr s under the sub gaussian error model which implies an essential gap between statistical optimality and computational efficiency for submatrix localization

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

information theoretic bounds and phase transitions in clustering sparse pca and submatrix localization we study the problem of detecting a structured low rank signal matrix corrupted with additive gaussian noise this includes clustering in a gaussian mixture model sparse pca and submatrix localization each of these problems is conjectured to exhibit a sharp information theoretic threshold below which the signal is too weak for any algorithm to detect we derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these planted models and null models where the signal matrix is zero for sparse pca and submatrix localization we determine this threshold exactly in the limit where the number of blocks is large or the signal matrix is very sparse for the clustering problem our bounds differ by a factor of sqrt 2 when the number of clusters is large moreover our upper bounds show that for each of these problems there is a significant regime where reliable detection is information theoretically possible but where known algorithms such as pca fail completely since the spectrum of the observed matrix is uninformative this regime is analogous to the conjectured hard but detectable regime for community detection in sparse graphs

eca high dimensional elliptical component analysis in non gaussian distributions abstractwe present a robust alternative to principal component analysis pca called elliptical component analysis eca for analyzing high dimensional elliptically distributed data eca estimates the eigenspace of the covariance matrix of the elliptical data to cope with heavy tailed elliptical distributions a multivariate rank statistic is exploited at the model level we consider two settings either that the leading eigenvectors of the covariance matrix are nonsparse or that they are sparse methodologically we propose eca procedures for both nonsparse and sparse settings theoretically we provide both nonasymptotic and asymptotic analyses quantifying the theoretical performances of eca in the nonsparse setting we show that eca s performance is highly related to the effective rank of the covariance matrix in the sparse setting the results are twofold i we show that the sparse eca estimator based on a combinatoric program attains the optimal rate of convergence ii based on some recent d

low dimensional q tori in fpu lattices dynamics and localization properties abstract recent studies on the fermi pasta ulam fpu paradox like the theory of q breathers and the metastability scenario dealing mostly with the energy localization properties in the fpu space of normal modes q space motivated our first work on q tori in the fpu problem christodoulidi et al 2010 19 the q tori are low dimensional invariant tori hosting trajectories that present features relevant to the interpretation of fpu recurrences as well as the energy localization in q space the present paper is a continuation of our work in christodoulidi et al 2010 19 our new results are we extend a method of analytical computation of q tori using poincare lindstedt series from the to the fpu and we reach significantly higher expansion orders using an improved computer algebraic program we probe numerically the convergence properties as well as the level of precision of our computed series we develop an additional algorithm in order to systematically locate values of the incommensurable frequencies used as an input in the pl series construction of q tori corresponding to progressively higher values of the energy we generalize a proposition proved in christodoulidi et al 2010 19 regarding the so called sequence of propagation of an initial excitation in the pl series we show by concrete examples how the latter interprets the localization patterns found in numerical simulations we focus in particular on various types of extensive initial excitations that lead to q tori solutions with exponentially localized profiles finally we discuss the relation between q tori q breathers viewed as one dimensional q tori and the so called fpu trajectories invoked in the original study of the fpu problem

a displacement based finite element formulation for general polyhedra using harmonic shape functions summary a displacement based continuous galerkin finite element formulation for general polyhedra is presented for applications in nonlinear solid mechanics the polyhedra can have an arbitrary number of vertices or faces the faces of the polyhedra can have an arbitrary number of edges and can be nonplanar the polyhedra can be nonconvex with only the mild restriction of star convexity with respect to the vertex averaged centroid conforming shape functions are constructed using harmonic functions defined on the undeformed configuration thus requiring the use of a total lagrangian finite element formulation in large deformation applications for nonlinear applications with computationally intensive constitutive models it is important to minimize the number of element quadrature points for this reason an integration scheme is adopted in which the number of quadrature points is equal to the number of vertices as a first step toward verifying the element behavior in the general nonlinear setting several linear verification examples are presented using both random voronoi meshes and distorted hexahedral meshes for the hexahedral meshes results for the polyhedral formulation are compared to those of the standard trilinear hexahedral formulation the element behavior in the nearly incompressible regime is also examined copyright 2013 john wiley sons ltd

estimation of functionals of sparse covariance matrices high dimensional statistical tests often ignore correlations to gain simplicity and stability leading to null distributions that depend on functionals of correlation matrices such as their frobenius norm and other l r norms motivated by the computation of critical values of such tests we investigate the difficulty of estimation the functionals of sparse correlation matrices specifically we show that simple plug in procedures based on thresholded estimators of correlation matrices are sparsity adaptive and minimax optimal over a large class of correlation matrices akin to previous results on functional estimation the minimax rates exhibit an elbow phenomenon our results are further illustrated in simulated data as well as an empirical study of data arising in financial econometrics

on metastability in fpu we present an analytical study of the fermi pasta ulam fpu model with periodic boundary conditions we analyze the dynamics corresponding to initial data with one low frequency fourier mode excited we show that correspondingly a pair of kdv equations constitute the resonant normal form of the system we also use such a normal form in order to prove the existence of a metastability phenomenon more precisely we show that the time average of the modal energy spectrum rapidly attains a well defined distribution corresponding to a packet of low frequencies modes subsequently the distribution remains unchanged up to the time scales of validity of our approximation the phenomenon is controlled by the specific energy

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

adaptive thresholding for sparse covariance matrix estimation in this article we consider estimation of sparse covariance matrices and propose a thresholding procedure that is adaptive to the variability of individual entries the estimators are fully data driven and demonstrate excellent performance both theoretically and numerically it is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm in contrast the commonly used universal thresholding estimators are shown to be suboptimal over the same parameter spaces support recovery is discussed as well the adaptive thresholding estimators are easy to implement the numerical performance of the estimators is studied using both simulated and real data simulation results demonstrate that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators the method is also illustrated in an analysis on a dataset from a small round blue cell tumor microarray experiment a supplement to this ar

energy localization on q tori long term stability and the interpretation of fermi pasta ulam recurrences we focus on two approaches that have been proposed in recent years for the explanation of the so called fermi pasta ulam fpu paradox i e the persistence of energy localization in the low q fourier modes of fermi pasta ulam nonlinear lattices preventing equipartition among all modes at low energies in the first approach a low frequency fraction of the spectrum is initially excited leading to the formation of natural packets exhibiting exponential stability while in the second emphasis is placed on the existence of q breathers i e periodic continuations of the linear modes of the lattice which are exponentially localized in fourier space following ideas of the latter we introduce in this paper the concept of q tori representing exponentially localized solutions on low dimensional tori and use their stability properties to reconcile these two approaches and provide a more complete explanation of the fpu paradox

minimax bounds for sparse pca with noisy high dimensional data we study the problem of estimating the leading eigenvectors of a high dimensional population covariance matrix based on independent gaussian observations we establish a lower bound on the minimax risk of estimators under the l2 loss in the joint limit as dimension and sample size increase to infinity under various models of sparsity for the population eigenvectors the lower bound on the risk points to the existence of different regimes of sparsity of the eigenvectors we also propose a new method for estimating the eigenvectors by a two stage coordinate selection scheme

the spectral norm of random inner product kernel matrices we study an inner product kernel random matrix model whose empirical spectral distribution was shown by xiuyuan cheng and amit singer to converge to a deterministic measure in the large n and p limit we provide an interpretation of this limit measure as the additive free convolution of a semicircle law and a marchenko pastur law by comparing the tracial moments of this matrix to an additive deformation of a wigner matrix we establish that for odd kernel functions the spectral norm of this matrix converges almost surely to the edge of the limiting spectrum our study is motivated by the analysis of a covariance thresholding procedure for the statistical detection and estimation of sparse principal components and our results characterize the limit of the largest eigenvalue of the thresholded sample covariance matrix in the null setting

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

rate optimal denoising of simultaneously sparse and low rank matrices we study minimax rates for denoising simultaneously sparse and low rank matrices in high dimensions we show that an iterative thresholding algorithm achieves near optimal rates adaptively under mild conditions for a large class of loss functions numerical experiments on synthetic datasets also demonstrate the competitive performance of the proposed method

monomial cubature rules since stroud a compilation a bibliography of references to cubature rules which have appeared since the publication of stroud s book 1971 is presented the standard regions that are treated in this paper are the n cube the n simplex the n sphere and the entire space

symmetric gaussian quadrature formulae for tetrahedronal regions abstract quadrature formulae of degrees 2 to 6 are presented for the numerical integration of a function over tetrahedronal regions the formulae presented are of gaussian type and fully symmetric with respect to the four vertices of the tetrahedron

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

a three dimensional unsplit godunov method for scalar conservation laws linear advection of a scalar quantity by a specified velocity field arises in a number of different applications of particular interest here is the transport of species and energy in low mach number models for combustion atmospheric flows and astrophysics as well as contaminant transport in darcy models of saturated subsurface flow an important characteristic of these problems is that the velocity field is not known analytically instead an auxiliary equation is solved to compute averages of the velocities over faces in a finite volume discretization in this paper we present a customized three dimensional finite volume advection scheme for this class of problems that provides accurate resolution for smooth problems while avoiding undershoot and overshoot for nonsmooth profiles the method is an extension of an algorithm by bell dawson and shubin bds which was developed for a class of scalar conservation laws arising in porous media flows in two dimensions the original bds algorithm is a variant of unsplit higher order godunov methods based on construction of a limited bilinear profile within each computational cell here we present a three dimensional extension of the original bds algorithm that is based on a limited trilinear profile within each cell we compare this new method to several other unsplit approaches including piecewise linear methods piecewise parabolic methods and wave propagation schemes

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

spatially adaptive stochastic methods for fluid structure interactions subject to thermal fluctuations in domains with complex geometries we develop stochastic mixed finite element methods for spatially adaptive simulations of fluid structure interactions when subject to thermal fluctuations to account for thermal fluctuations we introduce a discrete fluctuation dissipation balance condition to develop compatible stochastic driving fields for our discretization we perform analysis that shows our condition is sufficient to ensure results consistent with statistical mechanics we show the gibbs boltzmann distribution is invariant under the stochastic dynamics of the semi discretization to generate efficiently the required stochastic driving fields we develop a gibbs sampler based on iterative methods and multigrid to generate fields with o n computational complexity our stochastic methods provide an alternative to uniform discretizations on periodic domains that rely on fast fourier transforms to demonstrate in practice our stochastic computational methods we investigate within channel geometries having internal obstacles and no slip walls how the mobility diffusivity of particles depends on location our methods extend the applicability of fluctuating hydrodynamic approaches by allowing for spatially adaptive resolution of the mechanics and for domains that have complex geometries relevant in many applications

resolution adaptive fixed rank kriging abstractthe spatial random effects model is flexible in modeling spatial covariance functions and is computationally efficient for spatial prediction via fixed rank kriging frk however the model depends on a class of basis functions which if not selected properly may result in unstable or undesirable results additionally the maximum likelihood ml estimates of the model parameters are commonly computed using an expectation maximization em algorithm which further limits its applicability when a large number of basis functions are required in this research we propose a class of basis functions extracted from thin plate splines the functions are ordered in terms of their degrees of smoothness with higher order functions corresponding to larger scale features and lower order ones corresponding to smaller scale details leading to a parsimonious representation of a nonstationary spatial covariance function with the number of basis functions playing the role of spatial resolution the proposed

sparse principal component analysis and iterative thresholding principal component analysis pca is a classical dimension reduction method which projects data onto the principal subspace spanned by the leading eigenvectors of the covariance matrix however it behaves poorly when the number of features p is comparable to or even much larger than the sample size n in this paper we propose a new iterative thresholding approach for estimating principal subspaces in the setting where the leading eigenvectors are sparse under a spiked covariance model we find that the new approach recovers the principal subspace and leading eigenvectors consistently and even optimally in a range of high dimensional sparse settings simulated examples also demonstrate its competitive performance

signal detection in high dimension the multispiked case this paper applies le cam s asymptotic theory of statistical experiments to the signal detection problem in high dimension we consider the problem of testing the null hypothesis of sphericity of a high dimensional covariance matrix against an alternative of unspecified multiple symmetry breaking directions multispiked alternatives simple analytical expressions for the gaussian asymptotic power envelope and the asymptotic powers of previously proposed tests are derived those asymptotic powers remain valid for non gaussian data satisfying mild moment restrictions they appear to lie very substantially below the gaussian power envelope at least for small values of the number of symmetry breaking directions in contrast the asymptotic power of gaussian likelihood ratio tests based on the eigenvalues of the sample covariance matrix are shown to be very close to the envelope although based on gaussian likelihoods those tests remain valid under non gaussian densities satisfying mild moment conditions the results of this paper extend to the case of multispiked alternatives and possibly non gaussian densities the findings of an earlier study ann statist 41 2013 1204 1231 of the single spiked case the methods we are using here however are entirely new as the laplace approximation methods considered in the single spiked context do not extend to the multispiked case

optimal rates of convergence for estimating toeplitz covariance matrices toeplitz covariance matrices are used in the analysis of stationary stochastic processes and a wide range of applications including radar imaging target detection speech recognition and communications systems in this paper we consider optimal estimation of large toeplitz covariance matrices and establish the minimax rate of convergence for two commonly used parameter spaces under the spectral norm the properties of the tapering and banding estimators are studied in detail and are used to obtain the minimax upper bound the results also reveal a fundamental difference between the tapering and banding estimators over certain parameter spaces the minimax lower bound is derived through a novel construction of a more informative experiment for which the minimax lower bound is obtained through an equivalent gaussian scale model and through a careful selection of a finite collection of least favorable parameters in addition optimal rate of convergence for estimating the inverse of a toeplitz covariance matrix is also established

minimax sparse principal subspace estimation in high dimensions we study sparse principal components analysis in high dimensions where p the number of variables can be much larger than n the number of observations and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix we prove optimal non asymptotic lower and upper bounds on the minimax subspace estimation error under two different but related notions of lq subspace sparsity for 0 q 1 our upper bounds apply to general classes of covariance matrices and they show that lq constrained estimates can achieve optimal minimax rates without restrictive spiked covariance conditions 1 introduction principal components analysis pca was introduced in the early 20th century pearson 1901 hotelling 1933 and is arguably the most well known and widely used technique for dimension reduction it is part of the mainstream statistical repertoire and is routinely used in numerous and diverse areas of application however contemporary applications often involve much higher dimensional data than envisioned by the early developers of pca in such high dimensional situations where the number of variables p is of the same order or much larger than the number of observations n serious difficulties emerge standard pca can produce inconsistent estimates of the principal directions of variation and lead to unreliable conclusions johnstone and lu 2009 paul 2007 nadler 2008 the principal directions of variation correspond to the eigenvectors of the covariance matrix and in high dimensions consistent estimation of the eigenvectors is generally not possible without additional assumptions about the covariance matrix or its eigenstructure much of the recent development in pca has focused on methodology that applies the concept of sparsity to eigenvector estimation some examples include jolliffe trendafilov and uddin

asymptotics of the principal components estimator of large factor models with weakly influential factors abstract this paper introduces a drifting parameter asymptotic framework to derive accurate approximations to the finite sample distribution of the principal components pc estimator in situations when the factors explanatory power does not strongly dominate the explanatory power of the cross sectionally and temporally correlated idiosyncratic terms under our asymptotics the pc estimator is inconsistent we find explicit formulae for the amount of the inconsistency and propose an estimator of the number of factors for which the pc estimator works reasonably well for the special case when the idiosyncratic terms are cross sectionally but not temporally correlated or vice versa we show that the coefficients in the ols regressions of the pc estimates of factors loadings on the true factors true loadings are asymptotically normal and find explicit formulae for the corresponding asymptotic covariance matrix we explain how to estimate the parameters of the derived asymptotic distributions our monte carlo analysis suggests that our asymptotic formulae and estimators work well even for relatively small n and t we apply our theoretical results to test a hypothesis about the factor content of the us stock return data

minimax estimation in sparse canonical correlation analysis canonical correlation analysis is a widely used multivariate statistical technique for exploring the relation between two sets of variables this paper considers the problem of estimating the leading canonical correlation directions in high dimensional settings recently under the assumption that the leading canonical correlation directions are sparse various procedures have been proposed for many high dimensional applications involving massive data sets however there has been few theoretical justification available in the literature in this paper we establish rate optimal nonasymptotic minimax estimation with respect to an appropriate loss function for a wide range of model spaces two interesting phenomena are observed first the minimax rates are not affected by the presence of nuisance parameters namely the covariance matrices of the two sets of random variables though they need to be estimated in the canonical correlation analysis problem second we allow the presence of the residual canonical correlation directions however they do not influence the minimax rates under a mild condition on eigengap a generalized sin theta theorem and an empirical process bound for gaussian quadratic forms under rank constraint are used to establish the minimax upper bounds which may be of independent interest

3d block qt inversion of surface nuclear magnetic resonance data abstractsurface nuclear magnetic resonance surface nmr has up to now rarely been applied to 3d subsurface modeling inversion approaches currently in use are smooth inversion techniques that are not useful for identifying sharp geologic boundaries although they are already computationally expensive the resulting models are restricted to imaging the subsurface water content distribution and do not deliver relaxation times t2 based on the qt inversion scheme established elsewhere we have developed a method of 3d block qt inversion that uses horizontal smoothness constraints to resolve sharp boundaries in the vertical direction and the distributions of the water content and relaxation time t2 we have improved the computational efficiency i e the ability to perform the inversion using a common desktop computer by gating the surface nmr data reducing the model space to monoexponential decays within the subsurface bodies and inverting based on blocklike structures instead of smooth distributions