improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates the problem of estimating the eigenvalues and eigenvectors of the covariance matrix associated with a multivariate stochastic process is considered the focus is on finite sample size situations whereby the number of observations is limited and comparable in magnitude to the observation dimension using tools from random matrix theory and assuming a certain eigenvalue splitting condition new estimators of the eigenvalues and eigenvectors of the covariance matrix are derived that are shown to be consistent in a more general asymptotic setting than the traditional one indeed these estimators are proven to be consistent not only when the sample size increases without bound for a fixed observation dimension but also when the observation dimension increases to infinity at the same rate as the sample size numerical evaluations indicate that the estimators have an excellent performance in small sample size scenarios where the observation dimension and the sample size are comparable in magnitude

pointwise a posteriori error control for discontinuous galerkin methods for elliptic problems an a posteriori error bound for the maximum pointwise error for the interior penalty discontinuous galerkin method for a standard elliptic model problem on polyhedral domains is presented the computational domain is not required to be lipschitz thus allowing for domains with cracks and other irregular polyhedral domains the proof is based on the direct use of green s functions and varies substantially from the approach used in previous proofs of similar l infty estimates for continuous finite element methods in the literature numerical experiments indicating the good behavior of the resulting a posteriori bounds within an adaptive algorithm are also presented

testing independence among a large number of high dimensional random vectors capturing dependence among a large number of high dimensional random vectors is a very important and challenging problem by arranging n random vectors of length p in the form of a matrix we develop a linear spectral statistic of the constructed matrix to test whether the n random vectors are independent or not specifically the proposed statistic can also be applied to n random vectors each of whose elements can be written as either a linear stationary process or a linear combination of independent random variables the asymptotic distribution of the proposed test statistic is established for the case of as n to avoid estimating the spectrum of each random vector a modified test statistic which is based on splitting the original n vectors into two equal parts and eliminating the term that contains the inner structure of each random vector or time series is constructed the facts that the limiting distribution is normal and there is no need to know the inner structure of each investigated rando

clt for linear spectral statistics of large dimensional sample covariance matrices let bn 1 n t1 2nxnx nt1 2n where xn xij is n n with i i d complex standardized entries having finite fourth moment and t1 2n is a hermitian square root of the nonnegative definite hermitian matrix tn the limiting behavior as n with n n approaching a positive constant of functionals of the eigenvalues of bn where each is given equal weight is studied due to the limiting behavior of the empirical spectral distribution of bn it is known that these linear spectral statistics converges a s to a nonrandom quantity this paper shows their rate of convergence to be 1 n by proving after proper scaling that they form a tight sequence moreover if exppx211 0 and expp x11 4 2 or if x11 and tn are real and exppx411 3 they are shown to have gaussian limits

nonparametric estimate of spectral density functions of sample covariance matrices a first step the density function of the limiting spectral distribution of general sample covariance matrices is usually unknown we propose to use kernel estimators which are proved to be consistent a simulation study is also conducted to show the performance of the estimators

statistical eigen inference from large wishart matrices we consider settings where the observations are drawn from a zero mean multivariate real or complex normal distribution with the population covariance matrix having eigenvalues of arbitrary multiplicity we assume that the eigenvectors of the population covariance matrix are unknown and focus on inferential procedures that are based on the sample eigenvalues alone i e eigen inference results found in the literature establish the asymptotic normality of the fluctuation in the trace of powers of the sample covariance matrix we develop concrete algorithms for analytically computing the limiting quantities and the covariance of the fluctuations we exploit the asymptotic normality of the trace of powers of the sample covariance matrix to develop eigenvalue based procedures for testing and estimation specifically we formulate a simple test of hypotheses for the population eigenvalues and a technique for estimating the population eigenvalues in settings where the cumulative distribution function of the nonrandom population eigenvalues has a staircase structure monte carlo simulations are used to demonstrate the superiority of the proposed methodologies over classical techniques and the robustness of the proposed techniques in high dimensional relatively small sample size settings the improved performance results from the fact that the proposed inference procedures are global in a sense that we describe and exploit global information thereby overcoming the inherent biases that cripple classical inference procedures which are local and rely on local information

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

no eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices let b n 1 n t n 1 2 x n x n t n 1 2 where x n is n times n with i i d complex standardized entries having finite fourth moment and t n 1 2 is a hermitian square root of the nonnegative definite hermitian matrix t n it is known that as n to infty if n n converges to a positive number and the empirical distribution of the eigenvalues of t n converges to a proper probability distribution then the empirical distribution of the eigenvalues of b n converges a s to a nonrandom limit in this paper we prove that under certain conditions on the eigenvalues of t n for any closed interval outside the support of the limit with probability 1 there will be no eigenvalues in this interval for all n sufficiently large

on nondifferentiable functions and the bootstrap we investigate a class of statistical problems where usual bootstrap methods fail and discuss two alternative solutions in particular a stochastic procedure for constructing confidence sets is proposed special applications are the eigenvalues of a covariance matrix and minimum distance functionals

bootstrapping spectral statistics in high dimensions author s lopes me blandino a aue a abstract 2019 biometrika trust statistics derived from the eigenvalues of sample covariance matrices are called spectral statistics and they play a central role in multivariate testing although bootstrap methods are an established approach to approximating the laws of spectral statistics in low dimensional problems such methods are relatively unexplored in the high dimensional setting the aim of this article is to focus on linear spectral statistics as a class of prototypes for developing a new bootstrap in high dimensions a method we refer to as the spectral bootstrap in essence the proposed method originates from the parametric bootstrap and is motivated by the fact that in high dimensions it is difficult to obtain a nonparametric approximation to the full data generating distribution from a practical standpoint the method is easy to use and allows the user to circumvent the difficulties of complex asymptotic formulas for linear spectral statistics in addition to proving the consistency of the proposed method we present encouraging empirical results in a variety of settings lastly and perhaps most interestingly we show through simulations that the method can be applied successfully to statistics outside the class of linear spectral statistics such as the largest sample eigenvalue and others

fast exact bootstrap principal component analysis for p 1 million many have suggested a bootstrap procedure for estimating the sampling variability of principal component analysis pca results however when the number of measurements per subject p is much larger than the number of subjects n calculating and storing the leading principal components from each bootstrap sample can be computationally infeasible to address this we outline methods for fast exact calculation of bootstrap principal components eigenvalues and scores our methods leverage the fact that all bootstrap samples occupy the same n dimensional subspace as the original sample as a result all bootstrap principal components are limited to the same n dimensional subspace and can be efficiently represented by their low dimensional coordinates in that subspace several uncertainty metrics can be computed solely based on the bootstrap distribution of these low dimensional coordinates without calculating or storing the p dimensional bootstrap components fast bootstrap pca is applied to a dataset of sleep electroencephalogram recordings p 900 n 392 and to a dataset of brain magnetic resonance images mris p 3 million n 352 for the mri dataset our method allows for standard errors for the first 3 principal components based on 1000 bootstrap samples to be calculated on a standard laptop in 47 minutes as opposed to approximately 4 days with standard methods

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

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

adaptivity and blow up detection for nonlinear evolution problems this work is concerned with the development of a space time adaptive numerical method based on a rigorous a posteriori error bound for a semilinear convection diffusion problem which may exhibit blow up in finite time more specifically a posteriori error bounds are derived in the l infty l 2 l 2 h 1 type norm for a first order in time implicit explicit interior penalty discontinuous galerkin in space discretization of the problem although the theory presented is directly applicable to the case of conforming finite element approximations in space the choice of the discretization in time is made based on a careful analysis of adaptive time stepping methods for odes that exhibit finite time blow up the new adaptive algorithm is shown to accurately estimate the blow up time of a number of problems including one which exhibits regional blow up

minimal geodesics on groups of volume preserving maps and generalized solutions of the euler equations the three dimensional motion of an incompressible inviscid fluid is classically described by the euler equations but can also be seen following arnold 1 as a geodesic on a group of volume preserving maps local existence and uniqueness of minimal geodesics have been established by ebin and marsden 16 in the large for a large class of data the existence of minimal geodesics may fail as shown by shnirelman 26 for such data we show that the limits of approximate solutions are solutions of a suitable extension of the euler equations or equivalently are sharp measure valued solutions to the euler equations in the sense of diperna and majda 14 1999 john wiley sons inc

collapsing of riemannian manifolds and eigenvalues of laplace operator on etudie la perturbation des valeurs propres de l operateur de laplace sous des deformations des varietes qui ne conservent pas les dimensions

adaptive finite element methods for an optimal control problem involving dirac measures the purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so called pointwise tracking optimal control problem this linear quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the state thus leading to an adjoint problem with dirac measures on the right hand side control constraints are also considered the proposed error estimator relies on a posteriori error estimates in the maximum norm for the state and in muckenhoupt weighted sobolev spaces for the adjoint state we present an analysis that is valid for two and three dimensional domains we conclude by presenting several numerical experiments which reveal the competitive performance of adaptive methods based on the devised error estimator

an a posteriori error analysis of an elliptic optimal control problem in measure space abstract we propose an a posteriori error estimator for an optimal control problem with sparsity promoting cost the control variable lies in the space of regular borel measures we consider a solution technique that relies on the discretization of the control variable as a linear combination of dirac measures the proposed a posteriori error estimator can be decomposed into the sum of two contributions an error estimator in the maximum norm for the discretization of the adjoint equation and an estimator in the l 2 norm that accounts for the approximation of the state equation we prove that the designed error estimator is locally efficient and we explore its reliability properties the analysis is valid for two and three dimensional domains we illustrate the theory with numerical examples

a geometrical approach to monotone functions in r n this paper is concerned with the fine properties of monotone func tions on r n we study the continuity and differentiability properties of these functions the approximability properties the structure of the distributional derivatives and of the weak jacobians moreover we exhibit an example of a monotone function u which is the gradient of a c 1 convex function and

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

spectrum estimation for large dimensional covariance matrices using random matrix theory estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics the eigenvalues of covariance matrices play a key role in many widely used techniques in particular in principal component analysis pca in many modern data analysis problems statisticians are faced with large datasets where the sample size n is of the same order of magnitude as the number of variables p random matrix theory predicts that in this context the eigenvalues of the sample covariance matrix are not good estimators of the eigenvalues of the population covariance we propose to use a fundamental result in random matrix theory the marcenko pastur equation to better estimate the eigenvalues of large dimensional covariance matrices the marcenko pastur equation holds in very wide generality and under weak assumptions the estimator we obtain can be thought of as shrinking in a nonlinear fashion the eigenvalues of the sample covariance matrix to estimate the population eigenvalues inspired by ideas of random matrix theory we also suggest a change of point of view when thinking about estimation of high dimensional vectors we do not try to estimate directly the vectors but rather a probability measure that describes them we think this is a theoretically more fruitful way to think about these problems our estimator gives fast and good or very good results in extended simulations our algorithmic approach is based on convex optimization we also show that the proposed estimator is consistent

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

lower bounds on ricci curvature and the almost rigidity of warped products the basic rigidity theorems for manifolds of nonnegative or positive ricci curvature are the volume cone implies metric cone theorem the maximal diameter theorem cg and the splitting theorem cg each asserts that if a certain geometric quantity volume or diameter is as large as possible relative to the pertinent lower bound on ricci curvature then the metric on the manifold in question is a warped product metric of a particular type in this paper we provide quantitative generalizations of the above mentioned results among the applications are the splitting theorem for gromovhausdorff limit spaces x where mn x ricmn i see fy other applications include the assertion that for complete manifolds m with ricmf 0 and euclidean volume growth all tangent cones at infinity are metric cones compare bkn ct p1 via resealing arguments there are also strong consequences for the local structure of manifolds whose ricci curvature satisfies a fixed lower bound and for their gromov hausdorff limits some of these are announced in ccol for a more detailed discussion see cco2 cco3 cco4 our work further develops and significantly extends techniques which were introduced in col co2 and significantly extended in co3 in order to prove certain stability conjectures of anderson cheeger gromov and perelman the results of col co3 were announced in co4 we briefly review some of those results let dgh denote the gromov hausdorff distance between metric spaces see glp let s denote the unit sphere and recall that s is the unique complete

spectrum estimation from samples we consider the problem of approximating the set of eigenvalues of the covariance matrix of a multivariate distribution equivalently the problem of approximating the population spectrum given access to samples drawn from the distribution we consider this recovery problem in the regime where the sample size is comparable to or even sublinear in the dimensionality of the distribution first we propose a theoretically optimal and computationally efficient algorithm for recovering the moments of the eigenvalues of the population covariance matrix we then leverage this accurate moment recovery via a wasserstein distance argument to accurately reconstruct the vector of eigenvalues together this yields an eigenvalue reconstruction algorithm that is asymptotically consistent as the dimensionality of the distribution and sample size tend toward infinity even in the sublinear sample regime where the ratio of the sample size to the dimensionality tends to zero in addition to our theoretical results we show that our approach performs well in practice for a broad range of distributions and sample sizes

maximum norm a posteriori error estimates for singularly perturbed elliptic reaction diffusion problems residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polyhedral domains standard finite element approximations are considered the error constants are independent of the diameters of mesh elements and the small perturbation parameter in our analysis we employ sharp bounds on the green s function of the linearized differential operator numerical results are presented that support our theoretical findings