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

time varying autoregressive moving average models for covariance estimation we consider large scale covariance estimation using a small number of samples in applications where there is a natural ordering between the random variables the two classical approaches to this problem rely on banded covariance and banded inverse covariance structures corresponding to time varying moving average ma and autoregressive ar models respectively motivated by this analogy to spectral estimation and the well known modeling power of autoregressive moving average arma processes we propose a novel time varying arma covariance structure similarly to known results in the context of ar and ma we address the completion of an arma covariance matrix from its main band and its estimation based on random samples finally we examine the advantages of our proposed methods using numerical experiments

shrinking characteristics of precision matrix estimators we propose a framework to shrink a user specified characteristic of a precision matrix estimator that is needed to fit a predictive model estimators in our framework minimize the gaussian negative loglikelihood plus an l 1 penalty on a linear or affine function evaluated at the optimization variable corresponding to the precision matrix we establish convergence rate bounds for these estimators and propose an alternating direction method of multipliers algorithm for their computation our simulation studies show that our estimators can perform better than competitors when they are used to fit predictive models in particular we illustrate cases where our precision matrix estimators perform worse at estimating the population precision matrix but better at prediction

an operator valued t1 theory for symmetric czos abstract we provide a natural bmo criterion for the l 2 boundedness of calderon zygmund operators with operator valued kernels satisfying a symmetric property our arguments involve both classical and quantum probability theory in the appendix we give a proof of the l 2 boundedness of the commutators r j b whenever b belongs to the bourgain s vector valued bmo space where r j is the j th riesz transform a common ingredient is the operator valued haar multiplier studied by blasco and pott

variation inequalities for the fejer and poisson kernels in this paper we show that the g th order variation operator for both the fejer and poisson kernels are bounded from l p to l p 1 2 counterexamples are given if g 2

renormalization in quantum field theory an improved rigorous method the perturbative construction of the s matrix in the causal spacetime approach of epstein and glaser may be interpreted as a method of regularization for divergent feynman diagrams the results of any method of regularization must be equivalent to those obtained from the epstein glaser eg construction within the freedom left by the latter in particular the conceptually well defined approach of bogoliubov parasiuk hepp and zimmermann bphz though conceptually different from eg meets this requirement based on this equivalence we propose a modified bphz procedure which provides a significant simplification of the techniques of perturbation theory and which applies equally well to standard quantum field theory and to chiral theories we illustrate the proposed method by a number of examples of various orders in perturbation theory at the level of multi loop diagrams we confirm that subdiagrams as classified by zimmermann s forest formula in bphz can be restricted to subdiagrams in the sense of epstein glaser thus entailing an important reduction of actual computations the relationship of our approach to the method of dimensional regularization and renormalization is particularly transparent without having to invoke analytic continuation to unphysical spacetime dimension it sheds new light on the role of some parameters that appear within the dimensional regularization and thus establishes a direct link of this traditional method to the bphz scheme

on using a priori knowledge in space time adaptive processing in space time adaptive processing stap the clutter covariance matrix is routinely estimated from secondary ldquotarget freerdquo data because this type of data is more often than not rather scarce the so obtained estimates of the clutter covariance matrix are typically rather poor in knowledge aided ka stap an a priori guess of the clutter covariance matrix e g derived from knowledge of the terrain probed by the radar is available in this note we describe a computationally simple and fully automatic method for combining this prior guess with secondary data to obtain a theoretically optimal in the mean squared error sense estimate of the clutter covariance matrix the authors apply the proposed method to the kassper data set to illustrate the type of achievable performance

shrinkage to tapering estimation of large covariance matrices in this paper we introduce a shrinkage to tapering approach for estimating large covariance matrices when the number of samples is substantially fewer than the number of variables i e n p and p n the proposed estimator improves upon both shrinkage and tapering estimators by shrinking the sample covariance matrix to its tapered version we first show that under both normalized frobenius and spectral risks the minimum mean squared error mmse shrinkage to identity estimator is inconsistent and outperformed by a minimax tapering estimator for a class of high dimensional and diagonally dominant covariance matrices motivated by this observation we propose a shrinkage to tapering oracle sto estimator for efficient estimation of general large covariance matrices a closed form formula of the optimal coefficient of the proposed sto estimator is derived under the minimum frobenius risk since the true covariance matrix is to be estimated we further propose a sto approximating stoa algorithm with a data driven bandwidth selection procedure to iteratively estimate the coefficient and the covariance matrix we study the finite sample performances of different estimators and our simulation results clearly show the improved performances of the proposed sto estimators finally the proposed stoa method is applied to a real breast cancer gene expression data set

sparse linear discriminant analysis by thresholding for high dimensional data in many social economical biological and medical studies one objective is to classify a subject into one of several classes based on a set of variables observed from the subject because the probability distribution of the variables is usually unknown the rule of classification is constructed using a training sample the well known linear discriminant analysis lda works well for the situation where the number of variables used for classification is much smaller than the training sample size because of the advance in technologies modern statistical studies often face classification problems with the number of variables much larger than the sample size and the lda may perform poorly we explore when and why the lda has poor performance and propose a sparse lda that is asymptotically optimal under some sparsity conditions on the unknown parameters for illustration of application we discuss an example of classifying human cancer into two classes of leukemia based on a set of 7 129 genes and a training sample of size 72 a simulation is also conducted to check the performance of the proposed method

weighted variation inequalities for differential operators and singular integrals abstract we prove weighted strong q variation inequalities with 2 q for differential and singular integral operators for the first family of operators the weights used can be either sawyer s one sided a p weights or muckenhoupt s a p weights according to whether the differential operators in consideration are one sided or symmetric we use only muckenhoupt s a p weights for the second family all these inequalities hold equally in the vector valued case that is for functions with values in l for 1 as application we show variation inequalities for mean bounded positive invertible operators on l p with positive inverses

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

pointwise convergence for semigroups in vector valued l p spaces suppose that tt t 0 is a symmetric diffusion semigroup on l2 x and denote by widetilde t t t geq0 its tensor product extension to the bochner space l p x mathcal b where mathcal b belongs to a certain broad class of umd spaces we prove a vector valued version of the hopf dunford schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of widetilde t t t geq 0 on l p x mathcal b as an application we show that such continuations exhibit pointwise convergence

on the stress energy tensor of quantum fields in curved spacetimes comparison of different regularization schemes and symmetry of the hadamard seeley dewitt coefficients we review a few rigorous and partly unpublished results on the regularisation of the stress energy in quantum field theory on curved spacetimes 1 the symmetry of the hadamard seeley dewitt coefficients in smooth riemannian a lorentzian spacetimes 2 the equivalence of the local function and the hadamard point splitting procedure in smooth static spacetimes 3 the equivalence of the dewitt schwinger and the hadamard point splitting pro cedure in smooth riemannian and lorentzian spacetimes

nonlinear system theory another look at dependence based on the nonlinear system theory we introduce previously undescribed dependence measures for stationary causal processes our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems the proposed dependence measures provide a natural framework for a limit theory for stationary processes in particular under conditions with quite simple forms we present limit theorems for partial sums empirical processes and kernel density estimates the conditions are mild and easily verifiable because they are directly related to the data generating mechanisms

the strong p variation of martingales and orthogonal series let 1 p and letx x n n 0 be a sequence of scalars the strongp variation ofx denoted byw p x is defined as w p x sup left left x 0 p sum limits k 1 infty x n k x n k 1 p right 1 p right where the supremum runs over all increasing sequences of integers 0 n 0 n 1 n 2 let 1 p 2 and letm m n n 0 be a martingale inl p our main results are as follows if sigma mathbb e m n m n 1 p infty thenw p m is finite a s and we have mathbb e w p m p leqq c mathbb e m 0 p sum limits n geqq 1 mathbb e m n m n 1 p for some constantc depending only onp on the other hand let n be an arbitrary orthonormal system of functions inl 2 considerx x n n 0 inl 2 and lets n 0 x i i ands s n n 0 we prove that if x n p 1 p 2 thenw p s t for a e t and w p s 2 c x n p 1 p for some constantc each of these results is an extension of a result proved by bretagnolle for sums of independent mean zero r v s the casep 2 in also discussed our proofs use the real interpolation method of lions peetre they admit extensions in the banach space valued case provided suitable assumptions are imposed on the banach space

local causal structures hadamard states and the principle of local covariance in quantum field theory in the framework of the algebraic formulation we discuss and analyse some new features of the local structure of a real scalar quantum field theory in a strongly causal spacetime in particular we use the properties of the exponential map to set up a local version of a bulk to boundary correspondence the bulk is a suitable subset of a geodesic neighbourhood of an arbitrary but fixed point p of the underlying background while the boundary is a part of the future light cone having p as its own tip in this regime we provide a novel notion for the extended algebra of wick polynomials on the aforesaid cone and on the one hand we prove that it contains the information of the bulk counterpart via an injective homomorphism while on the other hand we associate to it a distinguished state whose pull back in the bulk is of hadamard form the main advantage of this point of view arises if one uses the universal properties of the exponential map and of the light cone in order to show that for any two given backgrounds m and m and for any two subsets of geodesic neighbourhoods of two arbitrary points it is possible to engineer the above procedure such that the boundary extended algebras are related via a restriction homomorphism this allows for the pull back of boundary states in both spacetimes and thus to set up a machinery which permits the comparison of expectation values of local field observables in m and m

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

regularized estimation in sparse high dimensional time series models many scientific and economic problems involve the analysis of high dimensional time series datasets however theoretical studies in high dimensional statistics to date rely primarily on the assumption of independent and identically distributed i i d samples in this work we focus on stable gaussian processes and investigate the theoretical properties of ell 1 regularized estimates in two important statistical problems in the context of high dimensional time series a stochastic regression with serially correlated errors and b transition matrix estimation in vector autoregressive var models we derive nonasymptotic upper bounds on the estimation errors of the regularized estimates and establish that consistent estimation under high dimensional scaling is possible via ell 1 regularization for a large class of stable processes under sparsity constraints a key technical contribution of the work is to introduce a measure of stability for stationary processes using their spectral properties that provides insight into the effect of dependence on the accuracy of the regularized estimates with this proposed stability measure we establish some useful deviation bounds for dependent data which can be used to study several important regularized estimates in a time series setting

shrinkage algorithms for mmse covariance estimation we address covariance estimation in the sense of minimum mean squared error mmse when the samples are gaussian distributed specifically we consider shrinkage methods which are suitable for high dimensional problems with a small number of samples large p small n first we improve on the ledoit wolf lw method by conditioning on a sufficient statistic by the rao blackwell theorem this yields a new estimator called rblw whose mean squared error dominates that of lw for gaussian variables second to further reduce the estimation error we propose an iterative approach which approximates the clairvoyant shrinkage estimator convergence of this iterative method is established and a closed form expression for the limit is determined which is referred to as the oracle approximating shrinkage oas estimator both rblw and oas estimators have simple expressions and are easily implemented although the two methods are developed from different perspectives their structure is identical up to specified constants the rblw estimator provably dominates the lw method for gaussian samples numerical simulations demonstrate that the oas approach can perform even better than rblw especially when n is much less than p we also demonstrate the performance of these techniques in the context of adaptive beamforming

regression coefficient and autoregressive order shrinkage and selection via the lasso summary the least absolute shrinkage and selection operator lasso has been widely used in regression shrinkage and selection we extend its application to the regression model with autoregressive errors two types of lasso estimators are carefully studied the first is similar to the traditional lasso estimator with only two tuning parameters one for regression coefficients and the other for autoregression coefficients these tuning parameters can be easily calculated via a data driven method but the resulting lasso estimator may not be fully efficient to overcome this limitation we propose a second lasso estimator which uses different tuning parameters for each coefficient we show that this modified lasso can produce the estimator as efficiently as the oracle moreover we propose an algorithm for tuning parameter estimates to obtain the modified lasso estimator simulation studies demonstrate that the modified estimator is superior to the traditional estimator one empirical example is also presented to illustrate the usefulness of lasso estimators the extension of the lasso to the autoregression with exogenous variables model is briefly discussed

covariance matrix estimation and classification with limited training data a new covariance matrix estimator useful for designing classifiers with limited training data is developed in experiments this estimator achieved higher classification accuracy than the sample covariance matrix and common covariance matrix estimates in about half of the experiments it achieved higher accuracy than regularized discriminant analysis but required much less computation

regularized estimation of linear functionals of precision matrices for high dimensional time series this paper studies a dantzig selector type regularized estimator for linear functionals of high dimensional linear processes explicit rates of convergence of the proposed estimator are obtained and they cover the broad regime from independent identically distributed samples to long range dependent time series and from sub gaussian innovations to those with mild polynomial moments it is shown that the convergence rates depend on the degree of temporal dependence and the moment conditions of the underlying linear processes the dantzig selector estimator is applied to the sparse markowitz portfolio allocation and the optimal linear prediction for time series in which the ratio consistency when compared with an oracle estimator is established the effect of dependence and innovation moment conditions is further illustrated in the simulation study finally the regularized estimator is applied to classify the cognitive states on a real functional magnetic resonance imaging dataset and to portfolio optimization on a financial dataset

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

weighted jump and variational inequalities for rough operators abstract in this paper we systematically study weighted jump and variational inequalities for rough operators more precisely we show some weighted jump and variational inequalities for the families t t e e 0 of truncated singular integrals and m m t t 0 of averaging operators with rough kernels which are defined respectively by t e f x y e y y n f x y d y and m t f x 1 t n y t y f x y d y where the kernel belongs to l q s n 1 for q 1

vector valued q variation for ergodic averages and analytic semigroups abstract in this paper we establish umd lattice valued variational inequalities for ergodic averages of contractively regular operators and analytic semigroups these results generalize on the one hand some scalar valued variational inequalities in ergodic theory on the other hand xu s very recent result on umd lattice valued maximal inequality as an application we deduce the jump estimates and obtain quantitative information on the rate of the pointwise convergence for semigroups acting on umd lattice valued functions

a direct approach to sparse discriminant analysis in ultra high dimensions summary sparse discriminant methods based on independence rules such as the nearest shrunken centroids classifier tibshirani et al 2002 and features annealed independence rules fan fan 2008 have been proposed as computationally attractive tools for feature selection and classification with high dimensional data a fundamental drawback of these rules is that they ignore correlations among features and thus could produce misleading feature selection and inferior classification we propose a new procedure for sparse discriminant analysis motivated by the least squares formulation of linear discriminant analysis to demonstrate our proposal we study the numerical and theoretical properties of discriminant analysis constructed via lasso penalized least squares our theory shows that the method proposed can consistently identify the subset of discriminative features contributing to the bayes rule and at the same time consistently estimate the bayes classification direction even when the dimension can grow faster than any polynomial order of the sample size the theory allows for general dependence among features simulated and real data examples show that lassoed discriminant analysis compares favourably with other popular sparse discriminant proposals

gaussian and bootstrap approximations for high dimensional u statistics and their applications this paper studies the gaussian and bootstrap approximations for the probabilities of a nondegenerate u statistic belonging to the hyperrectangles in mathbb r d when the dimension d is large a two step gaussian approximation procedure that does not impose structural assumptions on the data distribution is proposed subject to mild moment conditions on the kernel we establish the explicit rate of convergence uniformly in the class of all hyperrectangles in mathbb r d that decays polynomially in sample size for a high dimensional scaling limit where the dimension can be much larger than the sample size we also provide computable approximation methods for the quantiles of the maxima of centered u statistics specifically we provide a unified perspective for the empirical bootstrap the randomly reweighted bootstrap and the gaussian multiplier bootstrap with the jackknife estimator of covariance matrix as randomly reweighted quadratic forms and we establish their validity we show that all three methods are inferentially first order equivalent for high dimensional u statistics in the sense that they achieve the same uniform rate of convergence over all d dimensional hyperrectangles in particular they are asymptotically valid when the dimension d can be as large as o e n c for some constant c in 0 1 7 the bootstrap methods are applied to statistical applications for high dimensional non gaussian data including i principled and data dependent tuning parameter selection for regularized estimation of the covariance matrix and its related functionals ii simultaneous inference for the covariance and rank correlation matrices in particular for the thresholded covariance matrix estimator with the bootstrap selected tuning parameter we show that for a class of sub gaussian data error bounds of the bootstrapped thresholded covariance matrix estimator can be much tighter than those of the minimax estimator with a universal threshold in addition we also show that the gaussian like convergence rates can be achieved for heavy tailed data which are less conservative than those obtained by the bonferroni technique that ignores the dependency in the underlying data distribution

l_ 0 sparse inverse covariance estimation recently there has been focus on penalized log likelihood covariance estimation for sparse inverse covariance precision matrices the penalty is responsible for inducing sparsity and a very common choice is the convex l 1 norm however the best estimator performance is not always achieved with this penalty the most natural sparsity promoting norm is the nonconvex l 0 penalty but its lack of convexity has deterred its use in sparse maximum likelihood estimation in this paper we consider nonconvex l 0 penalized log likelihood inverse covariance estimation and present a novel cyclic descent algorithm for its optimization convergence to a local minimizer is proved which is highly nontrivial and we demonstrate via simulations the reduced bias and superior quality of the l 0 penalty as compared to the l 1 penalty

vector valued q variation for differential operators and semigroups i in this paper we establish mathcal b valued variational inequalities for differential operators ergodic averages and symmetric diffusion semigroups under the condition that banach space mathcal b has martingale cotype property these results generalize on the one hand pisier and xu s result on the variational inequalities for mathcal b valued martingales on the other hand many classical variational inequalities in harmonic analysis and ergodic theory moreover we show that rademacher cotype q is necessary for the mathcal b valued q variational inequalities as applications of the variational inequalities we deduce the jump estimates and obtain quantitative information on the rate of convergence it turns out the rate of convergence depends upon the geometry of the banach space under consideration while cowling and leinert have shown that convergence holds for all separable banach spaces

algebraic quantum field theory perturbation theory and the loop expansion the perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory we show that the algebra of interacting fields is additive i e fully determined by its subalgebras associated to arbitrary small subregions of minkowski space we also give an algebraic formulation of the loop expansion by introducing a projective system n of observables up to n loops where 0 is the poisson algebra of the classical field theory finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of green s functions

oscillation and variation for singular integrals in higher dimensions in this paper we continue our investigations of square function inequalities in harmonic analysis here we investigate oscillation and variation inequalities for singular integral operators in dimensions d 1 our estimates give quantitative information on the speed of convergence of truncations of a singular integral operator including upcrossing and a jump inequalities

steckin s theorem transference and spectral decompositions abstract let y be a closed subspace of lp where is an arbitrary measure and 1 sup v n n 0 1 2 in particular every surjective isometry of y can be expressed in the form v eia where a is well bounded of type b i e a has a spectral diagonalization analogous to though weaker than that in the spectral theorem for self adjoint operators this result which fails if y is replaced by an arbitrary reflexive space is obtained by a blend of the transference method of coifman and weiss with steckin s theorem and a recent result in abstract operator theory it has the direct consequence that every uniformly bounded one parameter group on y is the fourier stieltjes transform of a projection valued mapping of r an additional consequence is that every hermitian equivalent operator on y is well bounded of type b in the setting of an arbitrary banach space x power bounded operators with a logarithm of the form ia with a well bounded of type b are studied it is shown that if u is such an operator on x then for every function f of bounded variation on the unit circle n tf n u n converges in the strong operator topology this result which formally is a transference by u of steckin s theorem makes it possible to calculate directly from u a normalized logarithm for u and the spectral projections for the logarithm

conformal generally covariant quantum field theory the scalar field and its wick products in this paper we generalize the construction of generally covariant quantum theories given in bfv03 to encompass the conformal covariant case after introducing the abstract framework we discuss the massless conformally coupled klein gordon field theory showing that its quantization corresponds to a functor between two certain categories at the abstract level the ordinary fields could be thought of as natural transformations in the sense of category theory we show that the wick monomials without derivatives wick powers can be interpreted as fields in this generalized sense provided a non trivial choice of the renormalization constants is given a careful analysis shows that the transformation law of wick powers is characterized by a weight and it turns out that the sum of fields with different weights breaks the conformal covariance at this point there is a difference between the previously given picture due to the presence of a bigger group of covariance it is furthermore shown that the construction does not depend upon the scale appearing in the hadamard parametrix used to regularize the fields finally we briefly discuss some further examples of more involved fields

description of noncommutative theories and matrix models by wightman functions one of the main open problems of mathematical physics is to consistently quantize yang mills gauge theory if such a consistent quantization were to exist it is reasonable to expect a wightman reconstruction theorem by which a hilbert space and quantum field operators are recovered from n point functions however the original version of the wightman theorem is not equipped to deal with gauge fields or fields taking values in a noncommutative space this paper explores a generalization of the wightman construction which allows the fundamental fields to take values in an arbitrary topological algebra in particular the construction applies to fields valued in a lie algebra representation of the type required by yang mills theory this appears to be the correct framework for a generalized reconstruction theorem amenable to modern quantum theories such as gauge theories and matrix models we obtain the interesting result that a large class of quantum theories are expected to arise as limits of matrix

efficient minimax estimation of a class of high dimensional sparse precision matrices estimation of the covariance matrix and its inverse the precision matrix in high dimensional situations is of great interest in many applications in this paper we focus on the estimation of a class of sparse precision matrices which are assumed to be approximately inversely closed for the case that the dimensionality p can be much larger than the sample size n which is fundamentally different from the classical case that p n different in nature from state of the art methods that are based on penalized likelihood maximization or constrained error minimization based on the truncated neumann series representation we propose a computationally efficient precision matrix estimator that has a computational complexity of o p3 we prove that the proposed estimator is consistent in probability and in l2 under the spectral norm moreover its convergence is shown to be rate optimal in the sense of minimax risk we further prove that the proposed estimator is model selection consistent by establishing a convergence result under the entry wise norm simulations demonstrate the encouraging finite sample size performance and computational advantage of the proposed estimator the proposed estimator is also applied to a real breast cancer data and shown to outperform existing precision matrix estimators

dirac quantum fields on a manifold on globally hyperbolic lorentzian manifolds we construct field operators which satisfy the dirac equation and have a causal anticommutator ambiguities in the construction are removed by formulating the theory in terms of c algebras of local observables a generalized form of the haag kastler axioms is verified introduction a lorentzian manifold is a four dimensional manifold m together with a pseudo riemannian metric g of signature such manifolds are widely used as models for space time in particular in the absence of boundaries and gravitational fields one uses m r4 and g q minkowski metric which has components ab diag l 1 1 1 we are interested in formulating quantum field theories on general lorentzian manifolds this means giving up many ideas familiar from the usual minkowski space treatments thus there are no poincare transformations on m no vacuum no particle states and so forth all one is left with are the field equations these are to be solved taking as data a representation of the ccr or car over a space like hypersurface ccr canonical commutation relations car canonical anticommutation relations for linear equations this quantum problem can be reduced to a corresponding classical problem for which solutions can be constructed in favorable cases in the nonlinear case very singular formal solutions can be exhibited which one might hope to make rigorous the procedure sketched above is fraught with ambiguities what hypersurface should one take what representation of the ccr car and so forth in general there is no natural choice and one should show that all choices lead to the same theory to do this in a fundamental way it seems necessary to abandon the field operators themselves and instead consider the c algebras that they generate this algebraic approach was originally formulated for minkowski space and seems all the more appropriate here in the haag kastler approach 10 one focuses on the c algebras associated with local regions of space time the algebras of local observables the original formulation on minkowski space consisted of a list of structural properties or axioms which such algebras ought to obey in a recent paper 6 we have proposed a generalization of these axioms to lorentzian manifolds the scalar field was given as an example received by the editors july 23 1980 and in revised form december 11 1980 1980 mathematics subject classification primary 81a17 supported by national science foundation phy77 21740 1982 american mathematical society 0002 9947 82 0000 0 3 12 04 75

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

theory and application of covariance matrix tapers for robust adaptive beamforming we unify several seemingly disparate approaches to robust adaptive beamforming through the introduction of the concept of a covariance matrix taper cmt this is accomplished by recognizing that an important class of adapted pattern modification techniques are realized by the application of a conformal matrix taper to the original sample covariance matrix from the schur product theorem for positive semi definite matrices and kolmogorov s existence theorem we further establish that cmts are in fact the solution to a minimum variance optimum beamformer associated with an auxiliary stochastic process that is related to the original by a hadamard schur product this allows us to gain deeper insight into the design of both existing pattern modification techniques and new cmts that can for example simultaneously address several different design constraints such as pattern distortion due to insufficient sample support and weights mismatch due to nonstationary interference a new two dimensional 2 d cmt for space time adaptive radar applications designed to provide more robust clutter cancellation is also introduced since the cmt approach only involves a single matrix haddamard product it is also inherently low complexity the practical utility of the cmt approach is illustrated through its application to both spatial and spatio temporal adaptive beamforming examples

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

optimal feature selection in high dimensional discriminant analysis we consider the high dimensional discriminant analysis problem for this problem different methods have been proposed and justified by establishing exact convergence rates for the classification risk as well as the ell 2 convergence results to the discriminative rule however sharp theoretical analysis for the variable selection performance of these procedures have not been established even though model interpretation is of fundamental importance in scientific data analysis this paper bridges the gap by providing sharp sufficient conditions for consistent variable selection using the sparse discriminant analysis through careful analysis we establish rates of convergence that are significantly faster than the best known results and admit an optimal scaling of the sample size n dimensionality p and sparsity level s in the high dimensional setting sufficient conditions are complemented by the necessary information theoretic limits on the variable selection problem in the context of high dimensional discriminant analysis exploiting a numerical equivalence result our method also establish the optimal results for the road estimator and the sparse optimal scoring estimator furthermore we analyze an exhaustive search procedure whose performance serves as a benchmark and show that it is variable selection consistent under weaker conditions extensive simulations demonstrating the sharpness of the bounds are also provided

the epstein glaser approach to perturbative quantum field theory graphs and hopf algebras the paper aims at investigating perturbative quantum field theory in the approach of epstein and glaser eg and in particular its formulation in the language of graphs and hopf algebras has various has are encountered each one associated with a special combination of physical concepts such as normalization localization pseudounitarity causal regularization and renormalization the algebraic structures representing the perturbative expansion of the s matrix are imposed on operator valued distributions equipped with appropriate graph indices translation invariance ensures the algebras to be analytically well defined and graded total symmetry allows to formulate bialgebras the algebraic results are given embedded in the corresponding physical framework covering the two eg versions by fredenhagen and scharf that differ with respect to the concrete recursive implementation of causality besides the ultraviolet divergences occurring in feynman s representation are mathematically reasoned as a final result the change of the renormalization scheme in the context of eg is modeled via a ha and interpreted as the eg analog of kreimer s ha the paper aims at investigating perturbative quantum field theory in the approach of epstein and glaser eg and in particular its formulation in the language of graphs and hopf algebras has various has are encountered each one associated with a special combination of physical concepts such as normalization localization pseudounitarity causal regularization and renormalization the algebraic structures representing the perturbative expansion of the s matrix are imposed on operator valued distributions equipped with appropriate graph indices translation invariance ensures the algebras to be analytically well defined and graded total symmetry allows to formulate bialgebras the algebraic results are given embedded in the corresponding physical framework covering the two eg versions by fredenhagen and scharf that differ with respect to the concrete recursive implementation of causality besides the ultraviolet divergences occurring in feynman s representation are mathematically reasoned as a

comparison and anti concentration bounds for maxima of gaussian random vectors slepian and sudakov fernique type inequalities which compare expectations of maxima of gaussian random vectors under certain restrictions on the covariance matrices play an important role in probability theory especially in empirical process and extreme value theories here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of gaussian random vectors without any restriction on the covariance matrices we also establish an anti concentration inequality for the maximum of a gaussian random vector which derives a useful upper bound on the levy concentration function for the gaussian maximum the bound is dimension free and applies to vectors with arbitrary covariance matrices this anti concentration inequality plays a crucial role in establishing bounds on the kolmogorov distance between maxima of gaussian random vectors these results have immediate applications in mathematical statistics as an example of application we establish a conditional multiplier central limit theorem for maxima of sums of independent random vectors where the dimension of the vectors is possibly much larger than the sample size

a direct estimation approach to sparse linear discriminant analysis this article considers sparse linear discriminant analysis of high dimensional data in contrast to the existing methods which are based on separate estimation of the precision matrix and the difference of the mean vectors we introduce a simple and effective classifier by estimating the product directly through constrained l1 minimization the estimator can be implemented efficiently using linear programming and the resulting classifier is called the linear programming discriminant lpd rule the lpd rule is shown to have desirable theoretical and numerical properties it exploits the approximate sparsity of and as a consequence allows cases where it can still perform well even when and or cannot be estimated consistently asymptotic properties of the lpd rule are investigated and consistency and rate of convergence results are given the lpd classifier has superior finite sample performance and significant computational advantages over the existing methods that require separate estimation

on the marcenko pastur law for linear time series institute of mathematical statistics 2015 this paper is concerned with extensions of the classical ma enko pastur law to time series specifically p dimensional linear processes are considered which are built from innovation vectors with independent identically distributed real or complex valued entries possessing zero mean unit variance and finite fourth moments the coefficient matrices of the linear process are assumed to be simultaneously diagonalizable in this setting the limiting behavior of the empirical spectral distribution of both sample covariance and symmetrized sample autocovariance matrices is determined in the high dimensional setting p n c 0 for which dimension p and sample size n diverge to infinity at the same rate the results extend existing contributions available in the literature for the covariance case and are one of the first of their kind for the autocovariance case

complete determination of the singularity structure of zeta functions series of extended epstein type provide examples of non trivial zeta functions with important physical applications the regular part of their analytic continuation is seen to be a convergent or an asymptotic series their singularity structure is completely determined in terms of the wodzicki residue in its generalized form which is proven to yield the residua of all the poles of the zeta function and not just that of the rightmost pole obtainable from the dixmier trace the calculation is a very down to earth application of these powerful functional analytical methods in physics

on the issue of imposing boundary conditions on quantum fields an interesting example of the deep interrelation between physics and mathematics is obtained when trying to impose mathematical boundary conditions on physical quantum fields this procedure has recently been re examined with care comments on that and previous analysis are provided here together with considerations on the results of the purely mathematical zeta function method in an attempt at clarifying the issue hadamard regularization is invoked in order to fill the gap between the infinities appearing in the qft renormalized results and the finite values obtained in the literature with other procedures

epstein glaser renormalization and differential renormalization in the framework of causal perturbation theory by epstein and glaser the process of renormalization is precisely equivalent to the extension of time ordered distributions to coincident points this is achieved by a modified taylor subtraction on the corresponding test functions i show that the pullback of this operation to the distributions yields expressions known from differential renormalization the subtraction is equivalent to bphz subtraction in momentum space some examples from euclidean scalar field theory in flat and curved spacetime will be presented

the taylor lagrange scheme as a template for symmetry preserving renormalization procedures a general regularization renormalization scheme based on intrinsic properties of quantum fields as operator valued distributions with adequate test functions is presented the paracompactness property of the minkowskian or euclidean manifolds permits a unique definition of fields through integrals over the manifold based on test functions which are partition of unity pu these test functions turn out to provide a direct lorentz invariant scheme to the extension procedure of singular distributions and possess the unique property of being equal to their taylor remainder of any order when expressed through lagrange s formulae this remainder leads to specific procedures of extension in the uv and ir domains these results directly obtainable at the physical dimension d 4 are found to depend on an arbitrary scale present in the definition of any pu test functions and relevant to the final rg analysis of physical amplitudes the method is of general character in that it comprises the well known symmetry preserving procedures of bogoliubov parasiuk hepp zimmermann pauli villars subtractions at the level of propagators and dispersion relations symmetry preservation is indeed verified explicitly in simple qed qcd gauge boson contributions and in a covariant light front dynamics treatment of the yukawa model

high dimensionality effects in the markowitz problem and other quadratic programs with linear constraints risk underestimation we study the properties of solutions of quadratic programs with linear equality constraints whose parameters are estimated from data in the high dimensional setting where p the number of variables in the problem is of the same order of magnitude as n the number of observations used to estimate the parameters the markowitz problem in finance is a subcase of our study assuming normality and independence of the observations we relate the ecient frontier computed empirically to the true ecient frontier our computations show that there is a separation of the errors induced by estimating the mean of the observations and estimating the covariance matrix in particular the price paid for estimating the covariance matrix is an underestimation of the variance by a factor roughly equal to 1 p n therefore the risk of the optimal population solution is underestimated when we estimate it by solving a similar quadratic program with estimated parameters we also characterize the statistical behavior of linear functionals of the empirical optimal vector and show that they are biased estimators of the corresponding population quantities we investigate the robustness of our gaussian results by extending the study to certain elliptical models and models where our n observations are correlated in time we show a lack of robustness of the gaussian results but are still able to get results concerning rst order properties of the quantities of interest even in the case of relatively heavy tailed data we require two moments risk underestimation is still present in the elliptical case and more pronounced that in the gaussian case we discuss properties of the non parametric and parametric bootstrap in this context we show several results including the interesting fact that standard applications of the bootstrap generally yields inconsistent estimates of bias finally we propose some strategies to correct these problems and practically validate them in some simulations in all the paper we will assume that p n and n p tend to innity and p n

strong invariance principles for dependent random variables we establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds applications to linear and some nonlinear processes are discussed strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions

the microlocal spectrum condition and wick polynomials of free fields on curved spacetimes quantum fields propagating on a curved spacetime are investigated in terms of microlocal analysis we discuss a condition on the wave front set for the correspondingn point distributions called microlocal spectrum condition sc on minkowski space this condition is satisfied as a consequence of the usual spectrum condition based on radzikowski s determination of the wave front set of the two point function of a free scalar field satisfying the hadamard condition in the kay and wald sense we construct in the second part of this paper all wick polynomials including the energy momentum tensor for this field as operator valued distributions on the manifold and prove that they satisfy our microlocal spectrum condition