some experiments with simulated annealing for coloring graphs abstract methods of thermodynamical simulation have been used for several famous combinatorial optimization problems for graph coloring i e partition of the node set into as few independent sets as possible we describe a method of simulation such an approach is combined with other techniques for graph coloring experiments on random graphs show evidence that this combination gives better results than anyone of the original non combined methods

convergence rates of eigenvector empirical spectral distribution of large dimensional sample covariance matrix the eigenvector empirical spectral distribution vesd is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of covariance matrices in this paper we shall show that the kolmogorov distance between the expected vesd of sample covariance matrix and the mary cenko pastur distribution function is of order o n 1 2 given that data dimension n to sample size n ratio is bounded between 0 and 1 this convergence rate is established under finite 10th moment condition of the underlying distribution it is also shown that for any fixed 0 the convergence rates of vesd are o n 1 4 in probability and o n 1 4 almost surely requiring finite 8th moment of the underlying distribution

spectral functions of a symmetric linear relation with a directing mapping i the results of part i see 5 are applied to pairs of formally symmetric differential expressions to hermitian differential systems and to a reduced operator moment problem

neural networks for optimization problems with inequality constraints the knapsack problem a strategy for finding approximate solutions to discrete optimization problems with inequality constraints using mean field neural networks is presented the constraints x d 0 are encoded by x x terms in the energy function a careful treatment of the mean field approximation for the self coupling parts of the energy is crucial and results in an essentially parameter free algorithm this methodology is extensively tested on the knapsack problem of size up to 103 items the algorithm scales like nm for problems with n items and m constraints comparisons are made with an exact branch and bound algorithm when this is computationally possible n d 30 the quality of the neural network solutions consistently lies above 95 of the optimal ones at a significantly lower cpu expense for the larger problem sizes the algorithm is compared with simulated annealing and a modified linear programming approach for nonhomogeneous problems these produce good solutions whereas for the more difficult homogeneous problems the neural approach is a winner with respect to solution quality and or cpu time consumption the approach is of course also applicable to other problems of similar structure like set covering

tests for high dimensional covariance matrices we propose tests for sphericity and identity of high dimensional covariance matrices the tests are nonparametric without assuming a specific parametric distribution for the data they can accommodate situations where the data dimension is much larger than the sample size namely the large p small n situations we demonstrate by both theoretical and empirical studies that the tests have good properties for a wide range of dimensions and sample sizes we applied the proposed test on a microarray dataset on yorkshire gilts and tested for the covariance structure for the expression levels for sets of genes

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

a note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices recently johansson and johnstone proved that the distribution of the properly rescaled largest principal component of the complex real wishart matrix x x xtx converges to the tracy widom law as n p the dimensions of x tend to in some ratio n p 0 we extend these results in two directions first of all we prove that the joint distribution of the first second third etc eigenvalues of a wishart matrix converges after a proper rescaling to the tracy widom distribution second of all we explain how the combinatorial machinery developed for wigner random matrices in refs 27 38 and 39 allows to extend the results by johansson and johnstone to the case of x with non gaussian entries provided n p o p1 3 we also prove that max n1 2 p1 2 2 o p1 2 log p a e for general 0

an information based neural approach to generic constraint satisfaction a novel artificial neural network heuristic inn for general constraint satisfaction problems is presented extending a recently suggested method restricted to boolean variables in contrast to conventional ann methods it employs a particular type of non polynomial cost function based on the information balance between variables and constraints in a mean field setting implemented as an annealing algorithm the method is numerically explored on a testbed of graph coloring problems the performance is comparable to that of dedicated heuristics and clearly superior to that of conventional mean field annealing

universality of local eigenvalue statistics for some sample covariance matrices we consider random complex sample covariance matrices 1 x x where x is a p n random matrix with i i d entries of distribution it has been conjectured that both the distribution of the distance between nearest neighbor eigenva lues in the bulk and that of the smallest eigenvalues become in the limit n p n 1 the same as that identified for a complex gaussian distribution we prove these conjectures for a certain class of probability distributions c 2004 wiley periodicals inc

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

resolvent convergence and spectral approximations of sequences of self adjoint subspaces abstract this paper studies resolvent convergence and spectral approximations of sequences of self adjoint subspaces relations in complex hilbert spaces concepts of strong resolvent convergence norm resolvent convergence spectral inclusion and spectral exactness are introduced fundamental properties of resolvents of subspaces are studied by applying these properties several equivalent and sufficient conditions for convergence of sequences of self adjoint subspaces in the strong and norm resolvent senses are given it is shown that a sequence of self adjoint subspaces is spectrally inclusive under the strong resolvent convergence and spectrally exact under the norm resolvent convergence a sufficient condition is given for spectral exactness of a sequence of self adjoint subspaces in an open interval lacking essential spectral points in addition criteria are established for spectral inclusion and spectral exactness of a sequence of self adjoint subspaces that are defined on proper closed subspaces

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

concentration of haar measures with an application to random matrices abstract we show that the mixing times of random walks on compact groups can be used to obtain concentration inequalities for the respective haar measures as an application we derive a concentration inequality for the empirical distribution of eigenvalues of sums of random hermitian matrices with possible applications in free probability the advantage over existing techniques is that the new method can deal with functions that are non lipschitz or even discontinuous with respect to the usual metrics

an information based neural approach to constraint satisfaction a novel artificial neural network approach to constraint satisfaction problems is presented based on information theoretical considerations it differs from a conventional mean field approach in the form of the resulting free energy the method implemented as an annealing algorithm is numerically explored on a testbed of k sat problems the performance shows a dramatic improvement over that of a conventional mean field approach and is comparable to that of a state of the art dedicated heuristic gsat walk the real strength of the method however lies in its generality with minor modifications it is applicable to arbitrary types of discrete constraint satisfaction problems

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

asymptotic power of sphericity tests for high dimensional data this paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity we establish the convergence under the null hypothesis and the alternative of the log ratio of the joint densities of the sample covariance eigenvalues to a gaussian process indexed by the norm of the perturbation when the perturbation norm is larger than the phase transition threshold studied in baik et al 2005 the limiting process is degenerate and discrimination between the null and the alternative is asymptotically certain when the norm is below the threshold the process is non degenerate so that the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous using the asymptotic theory of statistical experiments we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region in particular we show that the asymptotic power of the tracy widom type tests is trivial whereas that of the eigenvalue based likelihood ratio test is strictly larger than the size and close to the power envelope

spectral rigidity for addition of random matrices at the regular edge abstract we consider the sum of two large hermitian matrices a and b with a haar unitary conjugation bringing them into a general relative position we prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of a and b as the dimension of the matrix increases this implies optimal rigidity of the eigenvalues and optimal rate of convergence in voiculescu s theorem our previous works 4 5 established these results in the bulk spectrum the current paper completely settles the problem at the spectral edges provided they have the typical square root behavior the key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge our results also hold if the haar unitary matrix is replaced by the haar orthogonal matrix

optimal bounds for convergence of expected spectral distributions to the semi circular law let mathbf x x jk j k 1 n denote a hermitian random matrix with entries x jk which are independent for 1 le j le k le n we consider the rate of convergence of the empirical spectral distribution function of the matrix mathbf x to the semi circular law assuming that mathbf e x jk 0 mathbf e x jk 2 1 and that begin aligned sup n ge 1 sup 1 le j k le n mathbf e x jk 4 mu 4 infty end aligned and begin aligned sup 1 le j k le n x jk le d 0n frac 1 4 end aligned by means of a recursion argument it is shown that the kolmogorov distance between the expected spectral distribution of the wigner matrix mathbf w frac 1 sqrt n mathbf x and the semicircular law is of order o n 1

weak convergence and rate of convergence of mimo capacity random variable recent works on the distribution function of the capacity of independent and identically distributed i i d and semicorrelated narrowband channels show that the outage capacity computed using a gaussian approximation is close to the true outage capacity even for few antennas motivated by physical scattering considerations we study a multi antenna channel model with independent entries that are not necessarily identically distributed and show the weak convergence of capacity to a gaussian random variable the channel model considered in this paper subsumes well studied cases like the i i d and separable correlation models and thus we generalize previous results on weak convergence of multi antenna capacity using recent results from random matrix theory we also study the rate of convergence of ergodic capacity of i i d channels to its limit value employing a well known conjecture from random matrix theory we establish tight results for the rate of convergence and show a dependence of this rate on signal to noise ratio

rank 1 real wishart spiked model in this paper we consider n dimensional real wishart matrices y in the class input amssym w bbb r sigma m in which all but one eigenvalue of is 1 let the nontrivial eigenvalue of be 1 then as n m with m n 2 finite and nonzero the eigenvalue distribution of y will converge into the marchenko pastur distribution inside a bulk region when increases from 0 one starts to see a stray eigenvalue of y outside of the support of the marchenko pastur density as this stray eigenvalue leaves the bulk region a phase transition will occur in the largest eigenvalue distribution of the wishart matrix in this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occurs we will first establish the results that are valid for all n and m and will use them to carry out the asymptotic analysis in particular we have derived a contour integral formula for the harish chandra itzykson zuber integral when x and y are real symmetric and y is a rank 1 matrix this allows us to write down a fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques as a result we obtain an integral formula for the largest eigenvalue distribution in the large n limit characterized by painleve transcendents the approach used in this paper is very different from a recent paper by bloemenal and virag in which the largest eigenvalue distribution was obtained using a stochastic operator method in particular the painleve formula for the largest eigenvalue distribution obtained in this paper is new 2012 wiley periodicals inc

non parametric detection of the number of signals hypothesis testing and random matrix theory detection of the number of signals embedded in noise is a fundamental problem in signal and array processing this paper focuses on the non parametric setting where no knowledge of the array manifold is assumed first we present a detailed statistical analysis of this problem including an analysis of the signal strength required for detection with high probability and the form of the optimal detection test under certain conditions where such a test exists second combining this analysis with recent results from random matrix theory we present a new algorithm for detection of the number of sources via a sequence of hypothesis tests we theoretically analyze the consistency and detection performance of the proposed algorithm showing its superiority compared to the standard minimum description length mdl based estimator a series of simulations confirm our theoretical analysis

regular approximations of spectra of singular discrete linear hamiltonian systems with one singular endpoint abstract this paper is concerned with regular approximations of spectra of singular discrete linear hamiltonian systems with one singular endpoint for any given self adjoint subspace extension sse of the corresponding minimal subspace its spectrum can be approximated by eigenvalues of a sequence of induced regular sses generated by the same difference expression on smaller finite intervals it is shown that every sse of the minimal subspace has a pure discrete spectrum and the k th eigenvalue of any given sse is exactly the limit of the k th eigenvalues of the induced regular sses that is spectral exactness holds in the limit circle case furthermore error estimates for the approximations of eigenvalues are given in this case in addition in the limit point and intermediate cases spectral inclusive holds

a note on eigenvalues of a class of singular continuous and discrete linear hamiltonian systems abstract in this paper we show that the analytic and geometric multiplicities of an eigenvalue of a class of singular linear hamiltonian systems are equal where both endpoints are in the limit circle cases the proof is fundamental and is given for both continuous and discrete hamiltonian systems the method used in this paper also works for both endpoints are regular or one endpoint is regular and the other is in the limit circle case

eigenvectors of some large sample covariance matrix ensembles we consider sample covariance matrices s n frac 1 p sigma n 1 2 x nx n sigma n 1 2 where xn is a n p real or complex matrix with i i d entries with finite 12th moment and n is a n n positive definite matrix in addition we assume that the spectral measure of n almost surely converges to some limiting probability distribution as n and p n 0 we quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type frac 1 n text tr g sigma n s n zi 1 where i is the identity matrix g is a bounded function and z is a complex number this is then used to compute the asymptotically optimal bias correction for sample eigenvalues paving the way for a new generation of improved estimators of the covariance matrix and its inverse

multiple hypergeometric functions probabilistic interpretations and statistical uses abstract this article reviews and interprets recent mathematics of special functions with emphasis on integral representations of multiple hypergeometric functions b c carlson s centrally important parameterized functions r and initially defined as dirichlet averages are expressed as probability generating functions of mixed multinomial distributions various nested families generalizing the dirichlet distributions are developed for bayesian inference in multinomial sampling and contingency tables in the case of many way tables this motivates a new generalization of the function these distributions are also useful for the modeling of populations of personal probabilities evolving under the process of inference from statistical data a remarkable new integral identity is adapted from carlson to represent the moments of quadratic forms under multivariate normal and more generally elliptically contoured distributions this permits the computation of such moments by simple quadrature

the largest eigenvalue of real symmetric hermitian and hermitian self dual random matrix models with rank one external source part i we consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric 1 hermitian 2 and hermitian self dual 4 random matrix models with rank 1 external source they are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues assuming the one band condition and certain regularities of the potential function we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value when the nonzero eigenvalue of the external source matrix is less than or equal to the critical value the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper in this paper we also give a definition of the external source model for all 0

perturbations of spectra of closed subspaces in banach spaces abstract this paper studies the variation of spectra of closed subspaces i e linear relations in banach spaces under some perturbations a concept of commutativity for subspaces in banach spaces is introduced and its properties are studied it is shown that boundedness of the inverse of a closed subspace is preserved under relatively bounded and small gap perturbations and perturbation about spectral condition separately by using these results upper semi continuity of spectra of closed subspaces is obtained and an error estimate of spectra of closed subspaces is given under bounded perturbation in the special case that the space is a hilbert space invariance of self adjointness of subspaces under relatively bounded and small gap perturbations and the variation of spectrum of a self adjoint subspace under bounded and hermitian perturbation are discussed the results obtained in the present paper generalize the corresponding results for closed operators to closed subspaces and some of which weaken certain assumptions of the related existing results

means of a dirichlet process and multiple hypergeometric functions the lauricella theory of multiple hypergeometric functions is used to shed some light on certain distributional properties of the mean of a dirichlet process this approach leads to several results which are illustrated here among these are a new and more direct procedure for determining the exact form of the distribution of the mean a correspondence between the distribution of the mean and the parameter of a dirichlet process a characterization of the family of cauchy distributions as the set of the fixed points of this correspondence and an extension of the markov krein identity moreover an expression of the characteristic function of the mean of a dirichlet process is obtained by resorting to an integral representation of a confluent form of the fourth lauricella function this expression is then employed to prove that the distribution of the mean of a dirichlet process is symmetric if and only if the parameter of the process is symmetric and to provide a new expression of the moment generating function of the variance of a dirichlet process

asymptotic performance of mmse receivers for large systems using random matrix theory random matrix theory is used to derive the limit and asymptotic distribution of signal to interference plus noise ratio sir for a class of suboptimal minimum mean square error mmse receivers applied to large random systems with unequal power users we prove that the limiting sir converges to a deterministic value when k and n go to infinity with lim k n y being a positive constant where k is the number of users and n is the number of degrees of freedom we also prove that the sir of each particular user is asymptotically gaussian for large n and derive the closed form expressions of the variance for the sir variable under real spreading and complex spreading channel environments it is revealed that for a given k n pair under certain mild conditions the variance of the sir for complex spreading channels is half of that for the corresponding real spreading channels since the suboptimal mmse receiver becomes optimal for the case when the users are equally powered our results show that the conjecture made by tse and zeitouni for the complex spreading case is not affirmative we also derive the asymptotic distribution for sir in decibels which provides better description when n is small numerical results and computer simulations are provided to evaluate the accuracy of various limiting and asymptotic results obtained in this paper

boundary eigenvalue problems for differential equations nη λpη with λ polynomial boundary conditions abstract the present paper deals with the spectral properties of boundary eigenvalue problems for differential equations of the form n p on a compact interval with boundary conditions which depend on the spectral parameter polynomially here n as well as p are regular differential operators of order n and p respectively with n p 0 the main results concern the completeness minimality and riesz basis properties of the corresponding eigenfunctions and associated functions they are obtained after a suitable linearization of the problem and by means of a detailed asymptotic analysis of the green s function the function spaces where the above properties hold are described by independent boundary conditions an application to a problem from elasticity theory is given

concentration and convergence rates for spectral measures of random matrices the topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest including in particular matrices drawn from haar measure on the classical lie groups random compressions of random hermitian matrices and the so called random sum of two independent random matrices in each case we estimate the expected wasserstein distance from the empirical spectral measure to a deterministic reference measure and prove a concentration result for that distance as a consequence we obtain almost sure convergence of the empirical spectral measures in all cases

large deviations for wigner s law and voiculescu s non commutative entropy we study the spectral measure of gaussian wigner s matrices and prove that it satisfies a large deviation principle we show that the good rate function which governs this principle achieves its minimum value at wigner s semicircular law which entails the convergence of the spectral measure to the semicircular law as a conclusion we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about voiculescu s non commutative entropy

stability of essential spectra of self adjoint subspaces under compact perturbations abstract this paper studies stability of essential spectra of self adjoint subspaces i e self adjoint linear relations under finite rank and compact perturbations in hilbert spaces relationships between compact perturbation of closed subspaces and relatively compact perturbation of their operator parts are first established this gives a characterization of compact perturbation in terms of difference between the operator parts of perturbed and unperturbed subspaces it is shown that a self adjoint subspace is still self adjoint under either relatively bounded perturbation with relative bound less than one or relatively compact perturbation or compact perturbation with a certain additional condition by using these results invariance of essential spectra of self adjoint subspaces is proved under relatively compact and compact perturbations separately as a special case finite rank perturbation is discussed the results obtained in this paper generalize the corresponding results for self adjoint operators to self adjoint subspaces

on the law of addition of random matrices normalized eigenvalue counting measure of the sum of two hermitian or real symmetric matrices a n and b n rotated independently with respect to each other by the random unitary or orthogonal haar distributed matrix u n i e a n u n b n u n is studied in the limit of large matrix order n convergence in probability to a limiting nonrandom measure is established a functional equation for the stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of a n and b n is obtained and studied

performance of statistical tests for single source detection using random matrix theory this paper introduces a unified framework for the detection of a single source with a sensor array in the context where the noise variance and the channel between the source and the sensors are unknown at the receiver the generalized maximum likelihood test is studied and yields the analysis of the ratio between the maximum eigenvalue of the sampled covariance matrix and its normalized trace using recent results from random matrix theory a practical way to evaluate the threshold and the p value of the test is provided in the asymptotic regime where the number k of sensors and the number n of observations per sensor are large but have the same order of magnitude the theoretical performance of the test is then analyzed in terms of receiver operating characteristic roc curve it is in particular proved that both type i and type ii error probabilities converge to zero exponentially as the dimensions increase at the same rate and closed form expressions are provided for the error exponents these theoretical results rely on a precise description of the large deviations of the largest eigenvalue of spiked random matrix models and establish that the presented test asymptotically outperforms the popular test based on the condition number of the sampled covariance matrix

optimization by simulated annealing an experimental evaluation part ii graph coloring and number partitioning this is the second in a series of three papers that empirically examine the competitiveness of simulated annealing in certain well studied domains of combinatorial optimization simulated annealing is a randomized technique proposed by s kirkpatrick c d gelatt and m p vecchi for improving local optimization algorithms here we report on experiments at adapting simulated annealing to graph coloring and number partitioning two problems for which local optimization had not previously been thought suitable for graph coloring we report on three simulated annealing schemes all of which can dominate traditional techniques for certain types of graphs at least when large amounts of computing time are available for number partitioning simulated annealing is not competitive with the differencing algorithm of n karmarkar and r m karp except on relatively small instances moreover if running time is taken into account natural annealing schemes cannot even outperform multiple random runs of the local optimization algorithms on which they are based in sharp contrast to the observed performance of annealing on other problems

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

universality of the edge distribution of eigenvalues of wigner random matrices with polynomially decaying distributions of entries we consider an ensemble of wigner symmetric random matrices a n a ij i j 1 n with matrix elements a ij being i i d symmetrically distributed random variables open image in new window we assume that open image in new window and that

convergence rates of the spectral distributions of large random quaternion self dual hermitian matrices in this paper convergence rates of the spectral distributions of quaternion self dual hermitian matrices are investigated we show that under conditions of finite 6th moments the expected spectral distribution of a large quaternion self dual hermitian matrix converges to the semicircular law in a rate of o n 1 2 and the spectral distribution itself converges to the semicircular law in rates o p n 2 5 and o a s n 2 5 eta those results include gse as a special case

tracy widom limit for the largest eigenvalue of a large class of complex sample covariance matrices we consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity more precisely let x be an n x p matrix and let its rows be i i d complex normal vectors with mean 0 and covariance p we show that for a large class of covariance matrices p the largest eigenvalue of x x is asymptotically distributed after recentering and rescaling as the tracy widom distribution that appears in the study of the gaussian unitary ensemble we give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance n and p the main theorem applies to a number of covariance models found in applications for example well behaved toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms under some conditions on the mass of the atoms are among the models the theorem can handle generalizations of the theorem to certain spiked versions of our models and a s results about the largest eigenvalue are given we also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics

some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size this paper analyzes whether standard covariance matrix tests work when dimensionality is large and in particular larger than sample size in the latter case the singularity of the sample covariance matrix makes likelihood ratio tests degenerate but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well defined we study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together with their ratio converging to a finite nonzero limit we find that the existing test for sphericity is robust against high dimensionality but not the test for equality of the covariance matrix to a given matrix for the latter test we develop a new correction to the existing test statistic that makes it robust against high dimensionality

a fourier view on the r transform and related asymptotics of spherical integrals abstract we estimate the asymptotics of spherical integrals of real symmetric or hermitian matrices when the rank of one matrix is much smaller than its dimension we show that it is given in terms of the r transform of the spectral measure of the full rank matrix and give a new proof of the fact that the r transform is additive under free convolution these asymptotics also extend to the case where one matrix has rank one but complex eigenvalue a result related with the analyticity of the corresponding spherical integrals

spectral properties of self adjoint subspaces abstract this paper focuses on spectral properties of self adjoint subspaces in the product space of a hilbert space several classifications of the spectrum of a self adjoint subspace are introduced including discrete spectrum essential spectrum continuous spectrum singular continuous spectrum absolutely continuous spectrum and singular spectrum their relationships between the subspace and its operator part are established respectively it is shown that all self adjoint subspace extensions of a closed hermitian subspace have the same essential spectrum as a simple application it is shown that every self adjoint subspace extension of the corresponding minimal subspace to a singular second order symmetric linear difference equation has a bounded spectrum under some conditions this shows that there is an essential difference between spectral properties of singular symmetric difference operators and classical regular or singular symmetric differential operators

phase transitions and the search problem we describe how techniques that were originally developed in statistical mechanics can be applied to search problems that arise commonly in artificial intelligence this approach is useful for understanding the typical behavior of classes of problems in particular these techniques predict that abrupt changes in computational cost analogous to physical phase transitions should occur universally as heuristic effectiveness or search space topology is varied we also present a number of open questions raised by these studies

regular approximations of spectra of singular second order symmetric linear difference equations abstract this paper is concerned with regular approximations of spectra of singular second order symmetric linear difference equations for any given self adjoint subspace extension of the corresponding minimal subspace its spectrum can be approximated by eigenvalues of a sequence of regular self adjoint subspace extensions generated by the same difference expression on smaller finite intervals

rate of convergence to the semi circular law a stochastic bound of order op n 1 2 for the kolmogorov distance between the spectral distribution function of an n n matrix from wigner ensemble and the distribution function of the semi circular law is obtained the result holds assuming that the twelfth moment of the entries of the matrix is uniformly bounded

complex singular wishart matrices and applications in this paper complex singular wishart matrices and their applications to information theory are investigated in particular a volume element on the space of positive semidefinite m x m hermitian matrices of rank n m is introduced and some transformation properties are established the jacobian for the change of variables in the singular value decomposition of general m x n complex matrices is derived then the density functions are formulated for all rank n complex singular wishart distributions from this the joint eigenvalue density of low rank complex wishart matrices are derived the derived densities are used to evaluate the most important information theoretic measure the so called ergodic channel capacity of multiple input multiple output mimo spatially correlated rayleigh distributed wireless communication channels

local stability of the free additive convolution abstract we prove that the system of subordination equations defining the free additive convolution of two probability measures is stable away from the edges of the support and blow up singularities by showing that the recent smoothness condition of kargin is always satisfied as an application we consider the local spectral statistics of the random matrix ensemble a u b u where u is a haar distributed random unitary or orthogonal matrix and a and b are deterministic matrices in the bulk regime we prove that the empirical spectral distribution of a u b u concentrates around the free additive convolution of the spectral distributions of a and b on scales down to n 2 3

local laws for polynomials of wigner matrices abstract we consider general self adjoint polynomials in several independent random matrices whose entries are centered and have the same variance we show that under certain conditions the local law holds up to the optimal scale i e the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory we prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i i d entries thus establishing the optimal bulk local law for these classes of ensembles in particular we generalize a similar result of anderson for anticommutator for more general polynomials our conditions are effectively checkable numerically

convergence rate for spectral distribution of addition of random matrices abstract let a and b be two n by n deterministic hermitian matrices and let u be an n by n haar distributed unitary matrix it is well known that the spectral distribution of the sum h a u b u converges weakly to the free additive convolution of the spectral distributions of a and b as n tends to infinity we establish the optimal convergence rate 1 n in the bulk of the spectrum

automatic pca dimension selection for high dimensional data and small sample sizes bayesian inference from high dimensional data involves the integration over a large number of model parameters accurate evaluation of such high dimensional integrals raises a unique set of issues these issues are illustrated using the exemplar of model selection for principal component analysis pca a bayesian model selection criterion based on a laplace approximation to the model evidence for determining the number of signal principal components present in a data set has previously been show to perform well on various test data sets using simulated data we show that for d dimensional data and small sample sizes n the accuracy of this model selection method is strongly affected by increasing values of d by taking proper account of the contribution to the evidence from the large number of model parameters we show that model selection accuracy is substantially improved the accuracy of the improved model evidence is studied in the asymptotic limit d at fixed ratio n d with 1 in this limit model selection based upon the improved model evidence agrees with a frequentist hypothesis testing approach

local law of addition of random matrices on optimal scale the eigenvalue distribution of the sum of two large hermitian matrices when one of them is conjugated by a haar distributed unitary matrix is asymptotically given by the free convolution of their spectral distributions we prove that this convergence also holds locally in the bulk of the spectrum down to the optimal scales larger than the eigenvalue spacing the corresponding eigenvectors are fully delocalized similar results hold for the sum of two real symmetric matrices when one is conjugated by haar orthogonal matrix

local spectral statistics of the addition of random matrices we consider the local statistics of h v x v u y u where v and u are independent haar distributed unitary matrices and x and y are deterministic real diagonal matrices in the bulk we prove that the gap statistics and correlation functions coincide with the gue in the limit when the matrix size n rightarrow infty under mild assumptions on x and certain rigidity assumptions on y the latter being an assumption on the convergence of the eigenvalues of y to the quantiles of its limiting spectral measure which we assume to have a density our method relies on running a carefully chosen diffusion on the unitary group and comparing the resulting eigenvalue process to dyson brownian motion our method also applies to the case when v and u are drawn from the orthogonal group our proof relies on the local law for h proved in bao et al commun math phys 349 3 947 990 2017 j funct anal 271 3 672 719 2016 adv math 319 251 291 2017 as well as the dbm convergence results of landon and yau commun math phys 355 3 949 1000 2017 and landon et al adv math 346 1137 1332 2019

the lebesgue decomposition of the free additive convolution of two probability distributions we prove that the free additive convolution of two borel probability measures supported on the real line can have a component that is singular continuous with respect to the lebesgue measure on mathbb r only if one of the two measures is a point mass the density of the absolutely continuous part with respect to the lebesgue measure is shown to be analytic wherever positive and finite the atoms of the free additive convolution of borel probability measures on the real line have been described by bercovici and voiculescu in a previous paper

laplace approximations for hypergeometric functions with matrix argument in this paper we present laplace approximations for two functions of matrix argument the type i confluent hypergeometric function and the gauss hypergeometric function both of these functions play an important role in distribution theory in multivariate analysis but from a practical point of view they have proved challenging and they have acquired a reputation for being difficult to approximate appealing features of the approximations we present are i they are fully explicit and simple to evaluate in practice and ii typically they have excellent numerical accuracy the excellent numerical accuracy is demonstrated in the calculation of noncentral moments of wilks a and the likelihood ratio statistic for testing block independence and in the calculation of the cdf of the noncentral distribution of wilks a via a sequential saddlepoint approximation relative error properties of these approximations are also studied and it is noted that the approximations have uniformly bounded relative errors in important cases

a concentration inequality and a local law for the sum of two random matrices let h n a n u n b n u n ast where an and bn are two n by n hermitian matrices and un is a haar distributed random unitary matrix and let mu h n mu a n mu b n be empirical measures of eigenvalues of matrices hn an and bn respectively then it is known see pastur and vasilchuk in commun math phys 214 249 286 2000 that for large n the measure mu h n is close to the free convolution of measures mu a n and mu b n where the free convolution is a non linear operation on probability measures the large deviations of the cumulative distribution function of mu h n from its expectation have been studied by chatterjee j funct anal 245 379 389 2007 in this paper we improve chatterjee s concentration inequality and show that it holds with the rate which is quadratic in n in addition we prove a local law for eigenvalues of h n n by showing that the normalized number of eigenvalues in an interval approaches the density of the free convolution of a and b provided that the interval has width log n 1 2

holder continuity of cumulative distribution functions for noncommutative polynomials under finite free fisher information abstract this paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory more precisely we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non microstates free fisher information highlighting the special case of lipschitz conjugate variables for the first time in this generality it is shown that the analytic distributions of those evaluations have holder continuous cumulative distribution functions with an explicit holder exponent that depends only on the degree of the considered polynomial for linear polynomials we reach in the case of finite non microstates free fisher information the optimal holder exponent 2 3 and get lipschitz continuity in the case of lipschitz conjugate variables in particular our results guarantee that such polynomial evaluations have finite logarithmic energy and thus finite non microstates free entropy which partially settles a conjecture of charlesworth and shlyakhtenko 8 we further provide a very general criterion that gives for weak approximations of measures having holder continuous cumulative distribution functions explicit rates of convergence in terms of the kolmogorov distance finally we combine these results to study the asymptotic eigenvalue distributions of polynomials in gues or matrices with more general gibbs laws for gibbs laws this extends the corresponding result obtained in 21 from convergence in distribution to convergence in kolmogorov distance in the gue case we even provide explicit rates which quantify results of 23 24 in terms of the kolmogorov distance

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized