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

covariance estimation in high dimensions via kronecker product expansions this paper presents a new method for estimating high dimensional covariance matrices the method permuted rank penalized least squares prls is based on a kronecker product series expansion of the true covariance matrix assuming an i i d gaussian random sample we establish high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity for covariance matrices of low separation rank our results establish that prls has significantly faster convergence than the standard sample covariance matrix scm estimator the convergence rate captures a fundamental tradeoff between estimation error and approximation error thus providing a scalable covariance estimation framework in terms of separation rank similar to low rank approximation of covariance matrices the mse convergence rates generalize the high dimensional rates recently obtained for the ml flip flop algorithm for kronecker product covariance estimation we show that a class of block toeplitz covariance matrices is approximatable by low separation rank and give bounds on the minimal separation rank r that ensures a given level of bias simulations are presented to validate the theoretical bounds as a real world application we illustrate the utility of the proposed kronecker covariance estimator for spatio temporal linear least squares prediction of multivariate wind speed measurements

large deviations for gibbs measures with singular hamiltonians and emergence of kahler einstein metrics in the present paper and the companion paper berman kahler einstein metrics canonical random point processes and birational geometry arxiv 1307 3634 2015 a probabilistic statistical mechanical approach to the construction of canonical metrics on complex algebraic varieties x is introduced by sampling temperature deformed determinantal point processes the main new ingredient is a large deviation principle for gibbs measures with singular hamiltonians which is proved in the present paper as an application we show that the unique kahler einstein metric with negative ricci curvature on a canonically polarized algebraic manifold x emerges in the many particle limit of the canonical point processes on x in the companion paper berman in 2015 the extension to algebraic varieties x with positive kodaira dimension is given and a conjectural picture relating negative temperature states to the existence problem for kahler einstein metrics with positive ricci curvature is developed

stable maps in higher dimensions we formulate a notion of stability for maps between polarised varieties which generalises kontsevich s definition when the domain is a curve and tian donaldson s definition of k stability when the target is a point we give some examples such as kodaira embeddings and fibrations we prove the existence of a projective moduli space of canonically polarised stable maps generalising the kontsevich alexeev moduli space of stable maps in dimensions one and two we also state an analogue of the yau tian donaldson conjecture in this setting relating stability of maps to the existence of certain canonical kahler metrics

k stability and parabolic stability parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces in this paper we show that the three notions of parabolic polystability k polystability and existence of constant scalar curvature kahler metrics on the iterated blowup are equivalent for certain polarizations close to the boundary of the kahler cone

convexity of the k energy on the space of kahler metrics and uniqueness of extremal metrics we establish the convexity of mabuchi s k energy functional along weak geodesics in the space of kahler potentials on a compact kahler manifold thus confirming a conjecture of chen and give some applications in kahler geometry including a proof of the uniqueness of constant scalar curvature metrics or more generally extremal metrics modulo automorphisms the key ingredient is a new local positivity property of weak solutions to the homogeneous monge ampere equation on a product domain whose proof uses plurisubharmonic variation of bergman kernels

matrix positivity preservers in fixed dimension i a classical theorem of i j schoenberg characterizes functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size obtaining similar characterizations in fixed dimension is intricate in this note we provide a solution to this problem in the polynomial case as consequences we derive tight linear matrix inequalities for hadamard powers of positive semidefinite matrices and a sharp asymptotic bound for the matrix cube problem involving hadamard powers

condition number regularized covariance estimation estimation of high dimensional covariance matrices is known to be a difficult problem has many applications and is of current interest to the larger statistics community in many applications including so called the large p small n setting the estimate of the covariance matrix is required to be not only invertible but also well conditioned although many regularization schemes attempt to do this none of them address the ill conditioning problem directly in this paper we propose a maximum likelihood approach with the direct goal of obtaining a well conditioned estimator no sparsity assumption on either the covariance matrix or its inverse are are imposed thus making our procedure more widely applicable we demonstrate that the proposed regularization scheme is computationally efficient yields a type of steinian shrinkage estimator and has a natural bayesian interpretation we investigate the theoretical properties of the regularized covariance estimator comprehensively including its regularization path and proceed to develop an approach that adaptively determines the level of regularization that is required finally we demonstrate the performance of the regularized estimator in decision theoretic comparisons and in the financial portfolio optimization setting the proposed approach has desirable properties and can serve as a competitive procedure especially when the sample size is small and when a well conditioned estimator is required

analysis of an epidemiological model structured by time since last infection abstract modeling time since last infection tsli provides a means of formulating epidemiological models with fewer state variables or epidemiological classes and more flexible descriptions of infectivity after infection and susceptibility after recovery than usual the model considered here has two time variables chronological time t and the tsli and it has only two classes never infected n and infected at least once i unlike most age structured epidemiological models in which the i equation is formulated using t i t ours uses a more general differential operator this allows weaker conditions for the infectivity and susceptibility functions and thus is more generally applicable we reformulate the model as an age dependent population problem for analysis so that published results for these types of problems can be applied including the existence and regularity of model solutions we also show how other coupled models having two types of time variables can be stated as age dependent population problems

hub discovery in partial correlation graphs one of the most important problems in large scale inference problems is the identification of variables that are highly dependent on several other variables when dependence is measured by partial correlations these variables identify those rows of the partial correlation matrix that have several entries with large magnitudes i e hubs in the associated partial correlation graph this paper develops theory and algorithms for discovering such hubs from a few observations of these variables we introduce a hub screening framework in which the user specifies both a minimum partial correlation rho and a minimum degree delta to screen the vertices the choice of rho and delta can be guided by our mathematical expressions for the phase transition correlation threshold rho c governing the average number of discoveries they can also be guided by our asymptotic expressions for familywise discovery rates under the assumption of large number p of variables fixed number n of multivariate samples and weak dependence under the null hypothesis that the dispersion covariance matrix is sparse these limiting expressions can be used to enforce familywise error constraints and to rank the discoveries in order of increasing statistical significance for n ll p the computational complexity of the proposed partial correlation screening method is low and is therefore highly scalable thus it can be applied to significantly larger problems than previous approaches the theory is applied to discovering hubs in a high dimensional gene microarray dataset

flexible covariance estimation in graphical gaussian models in this paper we propose a class of bayes estimators for the covariance matrix of graphical gaussian models markov with respect to a decomposable graph g working with the w pg family defined by letac and massam ann statist 35 2007 1278 1323 we derive closed form expressions for bayes estimators under the entropy and squared error losses the w pg family includes the classical inverse of the hyper inverse wishart but has many more shape parameters thus allowing for flexibility in differentially shrinking various parts of the covariance matrix moreover using this family avoids recourse to mcmc often infeasible in high dimensional problems we illustrate the performance of our estimators through a collection of numerical examples where we explore frequentist risk properties and the efficacy of graphs in the estimation of high dimensional covariance structures

two group infection age model including an application to nosocomial infection in this article we analyze the global asymptotic behavior of a two group si susceptible infected epidemic model with age of infection we prove that the model exhibits the traditional threshold behavior where the disease free equilibrium is globally asymptotically stable if the basic reproduction number is less than one and the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than one we conclude the paper by presenting an application to nosocomial infections moreover some numerical simulations are presented for this application

extremal kahler metrics on projectivized vector bundles we prove the existence of extremal nonconstant scalar curvature kahler metrics on certain unstable projectivized vector bundles p e m over a compact constant scalar curvature kahler manifold m with discrete holomorphic automorphism group in certain adiabatic kahler classes in particular the vector bundles e m are assumed to split as a direct sum of stable subbundles e el circle plus circle plus e s all having different mumford takemoto slope for example mu e 1 mu e s

on the convergence and singularities of the j flow with applications to the mabuchi energy the j flow of s k donaldson and x x chen is a parabolic flow on kahler manifolds with two kahler metrics it is the gradient flow of the j functional that appears in chen s formula for the mabuchi energy we find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the j flow to a critical metric we use this result to show that on manifolds with ample canonical bundle the mabuchi energy is proper on all kahler classes in an open neighborhood of the canonical class defined by a positivity condition this improves previous results of chen and of the second author we discuss the implications of this for the problem of the existence of constant scalar curvature kahler metrics we also study the singularities of the j flow and under certain conditions which always hold for dimension 2 derive some estimates away from a subvariety we discuss the conjectural remark of donaldson that if the j flow does not converge on a kahler surface then it should blow up over some curves of negative self intersection 2007 wiley periodicals inc

detection of gauss markov random fields with nearest neighbor dependency the problem of hypothesis testing against independence for a gauss markov random field gmrf is analyzed assuming an acyclic dependency graph an expression for the log likelihood ratio of detection is derived assuming random placement of nodes over a large region according to the poisson or uniform distribution and nearest neighbor dependency graph the error exponent of the neyman pearson detector is derived using large deviations theory the error exponent is expressed as a dependency graph functional and the limit is evaluated through a special law of large numbers for stabilizing graph functionals the exponent is analyzed for different values of the variance ratio and correlation it is found that a more correlated gmrf has a higher exponent at low values of the variance ratio whereas the situation is reversed at high values of the variance ratio

a thermodynamical formalism for monge ampere equations moser trudinger inequalities and kahler einstein metrics we develop a variational calculus for a certain free energy functional on the space of all probability measures on a kahler manifold x this functional can be seen as a generalization of mabuchi s k energy functional and its twisted versions to more singular situations applications to monge ampere equations of mean field type twisted kahler einstein metrics and moser trudinger type inequalities on miller manifolds are given tian s alpha invariant is generalized to singular measures allowing in particular a proof of the existence of kahler einstein metrics with positive ricci curvature that are singular along a given anti canonical divisor which combined with very recent developments concerning miller metrics with conical singularities confirms a recent conjecture of donaldson as another application we show that if the calabi flow in the anti canonical class exists for all times then it converges to a kahler einstein metric when a unique one exists which is in line with a well known conjecture c 2013 elsevier inc all rights reserved

blowing up and desingularizing constant scalar curvature kahler manifolds this paper is concerned with the existence of constant scalar curvature kahler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature kahler metrics we also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature kahler metrics

complete characterization of hadamard powers preserving loewner positivity monotonicity and convexity abstract entrywise powers of symmetric matrices preserving positivity monotonicity or convexity with respect to the loewner ordering arise in various applications and have received much attention recently in the literature following fitzgerald and horn 1977 8 it is well known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness similar phenomena have also recently been shown by hiai 2009 to occur for monotonicity and convexity in this paper we complete the characterization of all the entrywise powers below and above the critical exponents that are positive monotone or convex on the cone of positive semidefinite matrices we then extend the original problem by fully classifying the positive monotone or convex powers in a more general setting where additional rank constraints are imposed on the matrices we also classify the entrywise powers that are super sub additive with respect to the loewner ordering finally we extend all the previous characterizations to matrices with negative entries our analysis consequently allows us to answer a question raised by bhatia and elsner 2007 regarding the smallest dimension for which even extensions of the power functions do not preserve loewner positivity

traveling waves in a delayed sir epidemic model with nonlinear incidence we study the traveling waves of a diffusive sir epidemic model with a general nonlinear incidence we obtain the threshold dynamics for spatial spread of the disease we show that c is the minimum wave speed for the existence of traveling wave solutions we establish the existence and non existence of traveling wave solutions for a diffusive sir model with a general nonlinear incidence the existence proof is shown by introducing an auxiliary system applying schauder s fixed point theorem and then a limiting argument the nonexistence proof is obtained by two sided laplace transform when the speed is less than the critical velocity numerical simulations support the theoretical results we also point out the effects of the delay and the diffusion rate of the infective individuals on the spreading speed

invariant manifolds for volterra integral equations of convolution type abstract in this paper we develop some elements of a qualitative theory for nonlinear volterra integral equations of convolution type our starting point is a local semiflow associated with the equation and acting on a space of compactly supported forcing functions within that framework we discuss the variation of constants formula the saddle point property the center manifold and hopf bifurcation some equations from population biology get special attention

existence of twisted constant scalar curvature kahler metrics with a large twist suppose that there exist two kahler metrics omega and alpha such that the metric contraction of alpha with respect to omega is constant i e varlambda omega alpha text const we prove that for all large enough r 0 there exists a twisted constant scalar curvature kahler metric omega in the cohomology class omega satisfying s omega r varlambda omega alpha text const we discuss its implication to k stability and the continuity method recently proposed by chen

endemic models with arbitrarily distributed periods of infection i fundamental properties of the model this paper is dedicated to frank hoppensteadt on the occasion of his 60th birthday abstract a model is developed for the spread of anin fectious disease ina populationwith constant recruitment of new susceptibles and the fundamental properties of its solutions are analyzed the model allows for arbitrarily many stages of infection all of which have general length distributions and disease mortalities existence and uniqueness of solutions to the model equations are established a basic reproduction ratio is derived and related to the existence of an endemic equilibrium to the stability of the disease free equilibrium and to weak and strong endemicity persistence of the disease a characteristic equation is found the zeros of which determine the local stability of the endemic equilibrium and sufficient stability conditions are given for the case that infected individuals do not return into the susceptible class in a subsequent paper explicit sufficient and necessary stability conditions will be derived for the case that the disease dynamics are much faster than the demographics

an equivalent measure of partial correlation coefficients for high dimensional gaussian graphical models gaussian graphical models ggms are frequently used to explore networks such as gene regulatory networks among a set of variables under the classical theory of ggms the construction of gaussian graphical networks amounts to finding the pairs of variables with nonzero partial correlation coefficients however this is infeasible for high dimensional problems for which the number of variables is larger than the sample size in this article we propose a new measure of partial correlation coefficient which is evaluated with a reduced conditional set and thus feasible for high dimensional problems under the markov property and adjacency faithfulness conditions the new measure of partial correlation coefficient is equivalent to the true partial correlation coefficient in construction of gaussian graphical networks based on the new measure of partial correlation coefficient we propose a multiple hypothesis test based method for the construction of gaussian graphical networks furthermore we establish

covariance estimation in decomposable gaussian graphical models graphical models are a framework for representing and exploiting prior conditional independence structures within distributions using graphs in the gaussian case these models are directly related to the sparsity of the inverse covariance concentration matrix and allow for improved covariance estimation with lower computational complexity we consider concentration estimation with the mean squared error mse as the objective in a special type of model known as decomposable this model includes for example the well known banded structure and other cases encountered in practice our first contribution is the derivation and analysis of the minimum variance unbiased estimator mvue in decomposable graphical models we provide a simple closed form solution to the mvue and compare it with the classical maximum likelihood estimator mle in terms of performance and complexity next we extend the celebrated stein s unbiased risk estimate sure to graphical models using sure we prove that the mse of the mvue is always smaller or equal to that of the biased mle and that the mvue itself is dominated by other approaches in addition we propose the use of sure as a constructive mechanism for deriving new covariance estimators similarly to the classical mle all of our proposed estimators have simple closed form solutions but result in a significant reduction in mse

reproduction numbers and the stability of equilibria of si models for heterogeneous populations a major project in deterministic epidemiological modeling of heterogeneous populations is to find conditions for the local and global stability of the equilibria and to work out the relations among these stability conditions the thresholds for epidemic take off and endemicity and the basic reproduction number s most of the work to date has been on models of homogeneous populations of constant size motivated by their analysis of models of the dynamics of human immunodeficiency virus acquired immunodeficiency syndrome hiv aids the authors carry out this project for si models of diseases that have multiple stages which can lead to death and which infect heterogeneous populations with intricate mixing patterns and varying sizes in such models it is even difficult to find the analytical expression for the stability threshold and the reproduction number the authors show how the number of disease transmitting contacts by infectives can be used as a lyapunov function to carry out a systematic stabili

quickest detection for changes in maximal knn coherence of random matrices this paper addresses the problem of quickest detection of a change in the maximal coherence between columns of an n times p random matrix based on a sequence of matrix observations having a single unknown change point the random matrix is assumed to have identically distributed rows and the maximal coherence is defined as the largest of the p atopwithdelims 2 correlation coefficients associated with any row likewise the k nearest neighbor knn coherence is defined as the k th largest of these correlation coefficients the forms of the pre and postchange distributions of the observed matrices are assumed to belong to the family of elliptically contoured densities with sparse dispersion matrices but are otherwise unknown a nonparametric stopping rule is proposed that is based on the maximal knn sample coherence between columns of each observed random matrix this is a summary statistic that is related to a test of existence of a hub vertex in a sample correlation graph having degree at least k performance bounds on the delay and false alarm performance of the proposed stopping rule are obtained in the purely high dimensional regime where p rightarrow infty and n is fixed when the prechange dispersion matrix is diagonal it is shown that among all functions of the proposed summary statistic the proposed stopping rule is asymptotically optimal under a minimax quickest change detection model as the stopping threshold approaches infinity the theory developed also applies to sequential hypothesis testing and fixed sample size tests

product quantization for nearest neighbor search this paper introduces a product quantization based approach for approximate nearest neighbor search the idea is to decompose the space into a cartesian product of low dimensional subspaces and to quantize each subspace separately a vector is represented by a short code composed of its subspace quantization indices the euclidean distance between two vectors can be efficiently estimated from their codes an asymmetric version increases precision as it computes the approximate distance between a vector and a code experimental results show that our approach searches for nearest neighbors efficiently in particular in combination with an inverted file system results for sift and gist image descriptors show excellent search accuracy outperforming three state of the art approaches the scalability of our approach is validated on a data set of two billion vectors

construction of constant scalar curvature kahler cone metrics over a compact kahler manifold we provide a fredholm alternative result for the lichnerowicz operator associated to a kahler metric with conic singularities along a divisor we deduce several existence results of constant scalar curvature kahler metrics with conic singularities existence result under small deformations of kahler classes existence result over a fano manifold existence result over certain ruled manifolds in this last case we consider the projectivization of a parabolic stable holomorphic bundle this leads us to prove that the existing hermitian einstein metric on this bundle enjoys a regularity property along the divisor on the base

partial correlation estimation by joint sparse regression models this article features online supplementary material in this article we propose a computationally efficient approach space sparse partial correlation estimation for selecting nonzero partial correlations under the high dimension low sample size setting this method assumes the overall sparsity of the partial correlation matrix and employs sparse regression techniques for model fitting we illustrate the performance of space by extensive simulation studies it is shown that space performs well in both nonzero partial correlation selection and the identification of hub variables and also outperforms two existing methods we then apply space to a microarray breast cancer dataset and identify a set of hub genes that may provide important insights on genetic regulatory networks finally we prove that under a set of suitable assumptions the proposed procedure is asymptotically consistent in terms of model selection and parameter estimation

on the calabi flow in this paper we study the calabi flow on a polarized kahler manifold and some related problems we first give a precise statement on the short time existence of the calabi flow for any c 3 a m initial kahler potential as an application we prove a stability result any metric near a constant scalar curvature kahler csck metric will flow to a nearby csck metric exponentially fast secondly we prove that a compactness theorem in the space of the kahler metrics given uniform ricci bound and potential bound as an application we prove the calabi flow can be extended once ricci curvature stays uniformly bounded lastly we prove a removing singularity result about a weak constant scalar curvature metric in a punctured disc

detection of correlations with adaptive sensing the problem of detecting correlations from samples of a high dimensional gaussian vector has recently received a lot of attention in most existing work detection procedures are provided with a full sample however following common wisdom in experimental design the experimenter may have the capacity to make targeted measurements in an on line and adaptive manner in this paper we investigate such adaptive sensing procedures for detecting positive correlations it is shown that using the same number of measurements adaptive procedures are able to detect significantly weaker correlations than their nonadaptive counterparts we also establish minimax lower bounds that show the limitations of any procedure keywords sequential testing adaptive sensing high dimensional detection highdimensional detection sequential testing sparse covariance matrices sparse principal component analysis

extremal metrics on ruled manifolds abstract in this paper we consider a compact kahler manifold with extremal kahler metric and a mumford stable holomorphic bundle over it we proved that if the holomorphic vector field defining the extremal kahler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold then there exist extremal kahler metrics on the projectivization of the dual vector bundle

the calabi flow on kahler surfaces with bounded sobolev constant i we consider the formation of singularities along the calabi flow by assuming the uniformly bounded sobolev constants on kahler surfaces we prove that if curvature tensor is not uniformly bounded then one can form a singular model called deepest bubble such deepest bubble has to be a scalar flat ale kahler metric in certain kahler classes on toric fano surfaces the sobolev constants are a priori bounded along the calabi flow with small calabi energy we can also show in certain cases no deepest bubble can form along the flow it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently to illustrate our results more clearly we focus on an example on mathbb cp 2 blown up three points at generic position our result also implies existence of constant scalar curvature metrics on mathbb cp 2 blown up three points at generic position in the kahler classes where the exceptional divisors have the same area

large scale correlation screening this article addresses the problem of screening for variables with high correlations in high dimensional data in which there can be many fewer samples than variables we focus on threshold based correlation screening methods for three related applications screening for variables with large correlations within a single treatment autocorrelation screening screening for variables with large cross correlations over two treatments cross correlation screening and screening for variables that have persistently large autocorrelations over two treatments persistent correlation screening the novelty of correlation screening is that it identifies a smaller number of variables that are highly correlated with others compared with identifying a number of correlation parameters correlation screening suffers from a phase transition phenomenon as the correlation threshold decreases the number of discoveries increases abruptly we obtain asymptotic expressions for the mean number of discoveries and the phase t

global analysis of an infection age model with a class of nonlinear incidence rates abstract sir infection age models with a very general class of nonlinear incidence rates f s j are investigated we give a necessary and sufficient condition for global asymptotic stability of the free equilibrium related to the basic reproduction number furthermore additional conditions allow us to prove an exponential stability of this disease free equilibrium finally by using a lyapunov functional we show the global asymptotic stability of the endemic equilibrium whenever it exists

measures of uncertainty in market network analysis abstract a general approach to measure statistical uncertainty of different filtration techniques for market network analysis is proposed two measures of statistical uncertainty are introduced and discussed one is based on conditional risk for multiple decision statistical procedures and another one is based on average fraction of errors it is shown that for some important cases the second measure is a particular case of the first one the proposed approach is illustrated by numerical evaluation of statistical uncertainty for popular network structures minimum spanning tree planar maximally filtered graph market graph maximum cliques and maximum independent sets in the framework of gaussian network model of stock market