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

integrative linear discriminant analysis with guaranteed error rate improvement multiple types of data measured on a common set of subjects arise in many areas numerous empirical studies have found that integrative analysis of such data can result in better statistical performance in terms of prediction and feature selection however the advantages of integrative analysis have mostly been demonstrated empirically in the context of two class classification we propose an integrative linear discriminant analysis method and establish a theoretical guarantee that it achieves a smaller classification error than running linear discriminant analysis on each data type individually we address the issues of outliers and missing values frequently encountered in integrative analysis and illustrate our method through simulations and a neuroimaging study of alzheimer s disease

adaptive isogeometric analysis with hierarchical box splines abstract isogeometric analysis is a recently developed framework based on finite element analysis where the simple building blocks in geometry and solution space are replaced by more complex and geometrically oriented compounds box splines are an established tool to model complex geometry and form an intermediate approach between classical tensor product b splines and splines over triangulations local refinement can be achieved by considering hierarchically nested sequences of box spline spaces since box splines do not offer special elements to impose boundary conditions for the numerical solution of partial differential equations pdes we discuss a weak treatment of such boundary conditions along the domain boundary an appropriate domain strip is introduced to enforce the boundary conditions in a weak sense the thickness of the strip is adaptively defined in order to avoid unnecessary computations numerical examples show the optimal convergence rate of box splines and their hierarchical variants for the solution of pdes

approximations of groups characterizations of sofic groups and equations over groups we give new characterizations of sofic groups a group g is sofic if and only if it is a subgroup of a quotient of a direct product of alternating or symmetric groups a group g is sofic if and only if any system of equations solvable in all alternating groups is solvable over g the last characterization allows to express soficity of an existentially closed group by sentences

adaptively refined multi patch b splines with enhanced smoothness a spline space suitable for isogeometric analysis iga on multi patch domains is presented our construction is motivated by emerging requirements in isogeometric simulations in particular iga spaces should allow for adaptive mesh refinement and they should guarantee the optimal smoothness of the discretized solution even across interfaces of adjacent patches given a domain manifold m consisting of individual patches isomorphic to the unit square or cube that are glued together along interfaces we present a construction of multi patch b splines defined on them their smoothness is enhanced by locally modifying or merging basis functions around the boundary of each patch the resulting multi patch b splines with enhanced smoothness mpbes possess the property of local linear independence and form a non negative partition of unity moreover their span can be characterized as the linear space of all piecewise polynomial functions on the domain manifold that possess certain smoothness properties subsequently adaptively refined mpbes are obtained by generalizing the construction of truncated hierarchical th b splines more precisely a nested sequence of spaces spanned by mpbes is considered corresponding to steps of local enrichment in addition an inversely nested sequence of subdomains which are submanifolds of m is used to specify the local refinement level of functions in these spaces finally truncated hierarchical mpbes are obtained by means of the selection and truncation mechanism of thb splines the desired properties of linear independence and convex partition of unity are maintained the paper presents several numerical examples which demonstrate potential applications of the new basis in isogeometric analysis

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

regularization of wavelet approximations in this paper we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced the approach can apply readily to many other statistical contexts various new penalty functions are proposed the hard thresholding and soft thresholding estimators of donoho and johnstone are specific members of nonlinear regularized wavelet estimators they correspond to the lower and upper envelopes of a class of the penalized least squares estimators necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions the sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax to efficiently solve penalized least squares problems nonlinear regularized sobolev interpolators nrsi are proposed as initial estimators wh

effortless quasi interpolation in hierarchical spaces we present a general and simple procedure to construct quasi interpolants in hierarchical spaces such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement the proposed hierarchical quasi interpolants are described in terms of the so called truncated hierarchical basis assuming a quasi interpolant is selected for each space associated with a particular level in the hierarchy the hierarchical quasi interpolants are obtained without any additional manipulation the main properties like polynomial reproduction of the quasi interpolants selected at each level are locally preserved in the hierarchical construction we show how to construct hierarchical local projectors and the local approximation order of the underling hierarchical space is also investigated the presentation is detailed for the truncated hierarchical b spline basis and we discuss its extension to a more general framework

large covariance estimation through elliptical factor models we propose a general principal orthogonal complement thresholding poet framework for large scale covariance matrix estimation based on the approximate factor model a set of high level sufficient conditions for the procedure to achieve optimal rates of convergence under different matrix norms is established to better understand how poet works such a framework allows us to recover existing results for sub gaussian data in a more transparent way that only depends on the concentration properties of the sample covariance matrix as a new theoretical contribution for the first time such a framework allows us to exploit conditional sparsity covariance structure for the heavy tailed data in particular for the elliptical distribution we propose a robust estimator based on the marginal and spatial kendall s tau to satisfy these conditions in addition we study conditional graphical model under the same framework the technical tools developed in this paper are of general interest to high dimensional principal component analysis thorough numerical results are also provided to back up the developed theory

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

on the completeness of hierarchical tensor product b splines abstract given a grid in r d consisting of d bi infinite sequences of hyperplanes possibly with multiplicities orthogonal to the d axes of the coordinate system we consider the spaces of tensor product spline functions of a given degree on a multi cell domain such a domain consists of finite set of cells which are defined by the grid a piecewise polynomial function belongs to the spline space if its polynomial pieces on adjacent cells have a contact according to the multiplicity of the hyperplanes in the grid we prove that the connected components of the associated set of tensor product b splines whose support intersects the multi cell domain form a basis of this spline space more precisely if the intersection of the support of a tensor product b spline with the multi cell domain consists of several connected components then each of these components contributes one basis function in order to establish the connection to earlier results we also present further details relating to the three dimensional case with single knots only a hierarchical b spline basis is defined by specifying nested hierarchies of spline spaces and multi cell domains we adapt the techniques from giannelli and juttler 2013 to the more general setting and prove the completeness of this basis in the sense that its span contains all piecewise polynomial functions on the hierarchical grid with the smoothness specified by the grid and the degrees under certain assumptions on the domain hierarchy finally we introduce a decoupled version of the hierarchical spline basis that allows to relax the assumptions on the domain hierarchy in certain situations such as quadratic tensor product splines the decoupled basis provides the completeness property for any choice of the domain hierarchy

sub gaussian mean estimators we discuss the possibilities and limitations of estimating the mean of a real valued random variable from independent and identically distributed observations from a non asymptotic point of view in particular we define estimators with a sub gaussian behavior even for certain heavy tailed distributions we also prove various impossibility results for mean estimators

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

optimizing domain parameterization in isogeometric analysis based on powell sabin splines we address the problem of constructing a high quality parameterization of a given planar physical domain defined by means of a finite set of boundary curves we look for a geometry map represented in terms of powell sabin b splines powell sabin splines are c 1 quadratic splines defined on a triangulation and thus the parameter domain can be any polygon the geometry map is generated by the following three step procedure first the shape of the parameter domain and a corresponding triangulation are determined in such a way that its number of corners matches the number of corners of the physical domain second the boundary control points related to the powell sabin b spline representation are chosen so that they parameterize the boundary curve of the physical domain third the remaining inner control points are obtained by solving a nimble optimization problem based on the winslow functional the proposed domain parameterization procedure is illustrated numerically in the context of isogeometric galerkin discretizations based on powell sabin splines it turns out that the flexibility rising from the generality of the parameter domain has a beneficial effect on the quality of the parameterization and also on the accuracy of the computed approximate solution

diameters of finite simple groups sharp bounds and applications let g be a finite simple group and let s be a normal subset of g we determine the diameter of the cayley graph r g s associated with g and s up to a multiplicative constant many applications follow for example we deduce that there is a constant c such that every element of g is a product of c involutions and we generalize this to elements of arbitrary order we also show that for any word w w xl xd there is a constant c c w such that for any simple group g on which w does not vanish every element of g is a product of c values of w from this we deduce that every verbal subgroup of a semisimple profinite group is closed other applications concern covering numbers expanders and random walks on finite simple groups

bounded normal generation and invariant automatic continuity abstract we study the question of how quickly products of a fixed conjugacy class in the projective unitary group of a ii1 factor von neumann algebra cover the entire group our result is that the number of factors that are needed is essentially as small as permitted by the 1 norm in analogy to results of liebeck and shalev for non abelian finite simple groups as an application of the techniques we prove that every homomorphism from the projective unitary group of a finite factor to a polish sin group is continuous a result which is even new for pu n moreover we show that the projective unitary group of a ii1 factor carries a unique polish group topology

asymptotic normality and optimalities in estimation of large gaussian graphical models the gaussian graphical model a popular paradigm for studying relationship among variables in a wide range of applications has attracted great attention in recent years this paper considers a fundamental question when is it possible to estimate low dimensional parameters at parametric square root rate in a large gaussian graphical model a novel regression approach is proposed to obtain asymptotically efficient estimation of each entry of a precision matrix under a sparseness condition relative to the sample size when the precision matrix is not sufficiently sparse or equivalently the sample size is not sufficiently large a lower bound is established to show that it is no longer possible to achieve the parametric rate in the estimation of each entry this lower bound result which provides an answer to the delicate sample size question is established with a novel construction of a subset of sparse precision matrices in an application of le cam s lemma moreover the proposed estimator is proven to have optimal convergence rate when the parametric rate cannot be achieved under a minimal sample requirement the proposed estimator is applied to test the presence of an edge in the gaussian graphical model or to recover the support of the entire model to obtain adaptive rate optimal estimation of the entire precision matrix as measured by the matrix ell q operator norm and to make inference in latent variables in the graphical model all of this is achieved under a sparsity condition on the precision matrix and a side condition on the range of its spectrum this significantly relaxes the commonly imposed uniform signal strength condition on the precision matrix irrepresentability condition on the hessian tensor operator of the covariance matrix or the ell 1 constraint on the precision matrix numerical results confirm our theoretical findings the roc curve of the proposed algorithm asymptotic normal thresholding ant for support recovery significantly outperforms that of the popular glasso algorithm

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

from nurbs to nurps geometries abstract quadratic powell sabin splines and their rational extension the so called nurps surfaces are an interesting alternative for classical tensor product nurbs in the context of isogeometric analysis because they allow the use of local refinements while retaining a b spline like representation and exact description of conic sections in this paper we present a simple and effective strategy to convert a given planar geometry defined by a quadratic nurbs representation into a nurps representation suitable for the analysis

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

bandits with heavy tail the stochastic multiarmed bandit problem is well understood when the reward distributions are sub gaussian in this paper we examine the bandit problem under the weaker assumption that the distributions have moments of order 1 e for some e 0 1 surprisingly moments of order 2 i e finite variance are sufficient to obtain regret bounds of the same order as under sub gaussian reward distributions in order to achieve such regret we define sampling strategies based on refined estimators of the mean such as the truncated empirical mean catoni s m estimator and the median of means estimator we also derive matching lower bounds that also show that the best achievable regret deteriorates when e 1

hyperlinearity essentially free actions and l2 invariants the sofic property we prove that connes embedding conjecture holds for the von neumann algebras of sofic groups that is sofic groups are hyperlinear hence we provide some new examples of hyperlinearity we also show that the determinant conjecture holds for sofic groups as well we introduce the notion of essentially free actions and amenable actions and study their properties

robust estimation of high dimensional covariance and precision matrices high dimensional data are often most plausibly generated from distributions with complex structure and leptokurtosis in some or all components covariance and precision matrices provide a useful summary of such structure yet the performance of popular matrix estimators typically hinges upon a sub gaussianity assumption this paper presents robust matrix estimators whose performance is guaranteed for a much richer class of distributions the proposed estimators under a bounded fourth moment assumption achieve the same minimax convergence rates as do existing methods under a sub gaussianity assumption consistency of the proposed estimators is also established under the weak assumption of bounded 2 e moments for e 0 2 the associated convergence rates depend on e

generators and commutators in finite groups abstract quotients of compact groups the first part of the paper establishes results about products of commutators in a d generator finite group g for example if h g g1 gr then every element of the subgroup h g is a product of f r factors of the form h 1 g 1 h 1 prime g 1 1 ldots lbrack h r g r h r prime g r 1 with h 1 h 1 prime ldots allowbreak h r h r prime in h under certain conditions on h a similar conclusion holds with the significantly weaker hypothesis that g h g1 gr where f r is replaced by f1 d r the results are applied in the second part of the paper to the study of normal subgroups in finitely generated profinite groups and in more general compact groups results include the characterization of topologically finitely generated compact groups which have a countably infinite image and of those which have a virtually dense normal subgroup of infinite index as a corollary it is deduced that a compact group cannot have a finitely generated infinite abstract quotient

generalized thresholding of large covariance matrices we propose a new class of generalized thresholding operators that combine thresholding with shrinkage and study generalized thresholding of the sample covariance matrix in high dimensions generalized thresholding of the covariance matrix has good theoretical properties and carries almost no computational burden we obtain an explicit convergence rate in the operator norm that shows the tradeoff between the sparsity of the true model dimension and the sample size and shows that generalized thresholding is consistent over a large class of models as long as the dimension p and the sample size n satisfy log p n 0 in addition we show that generalized thresholding has the sparsistency property meaning it estimates true zeros as zeros with probability tending to 1 and under an additional mild condition is sign consistent for nonzero elements we show that generalized thresholding covers as special cases hard and soft thresholding smoothly clipped absolute deviation and adaptive lasso and comp

on the lattice of normal subgroups in ultraproducts of compact simple groups we prove that the lattice of normal subgroups of ultraproducts of compact simple non abelian groups is distributive in the case of ultraproducts of finite simple groups or compact connected simple lie groups of bounded rank the set of normal subgroups is shown to be linearly ordered by inclusion 2010 mathematics subject classification 03c20 20d30 primary 20f69 20d06 22e46 secondary

parameter free weak imposition of dirichlet boundary conditions and coupling of trimmed and non conforming patches we present a parameter free domain sewing approach for low order as well as high order finite elements its final form contains only primal unknowns that is the approach does not introduce additional unknowns at the interface additionally it does not involve problem dependent parameters which require an estimation the presented approach is symmetry preserving that is the resulting discrete form of an elliptic equation will remain symmetric and positive definite it preserves the order of the underlying discretization and we demonstrate high order accuracy for problems of non matching discretizations concerning the mesh size h as well as the polynomial degree of the order of discretization p we also demonstrate how the method may be used to model material interfaces which may be curved and for which the interface does not coincide with the underlying mesh this novel approach is presented in the context of the p version and b spline version of the finite cell method an embedded domain method of high order and compared with more classical methods such as the penalty method or nitsche s method