regularized tyler s scatter estimator existence uniqueness and algorithms this paper considers the regularized tyler s scatter estimator for elliptical distributions which has received considerable attention recently various types of shrinkage tyler s estimators have been proposed in the literature and proved work effectively in the large p small n scenario nevertheless the existence and uniqueness properties of the estimators are not thoroughly studied and in certain cases the algorithms may fail to converge in this work we provide a general result that analyzes the sufficient condition for the existence of a family of shrinkage tyler s estimators which quantitatively shows that regularization indeed reduces the number of required samples for estimation and the convergence of the algorithms for the estimators for two specific shrinkage tyler s estimators we also proved that the condition is necessary and the estimator is unique finally we show that the two estimators are actually equivalent numerical algorithms are also derived based on the majorization minimization framework under which the convergence is analyzed systematically

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

improved calibration of high dimensional precision matrices estimation of a precision matrix i e the inverse covariance matrix is a fundamental problem in statistical signal processing applications when the observation dimension is of the same order of magnitude as the number of samples the conventional estimators of covariance matrix and its inverse perform poorly in order to obtain well behaved estimators in high dimensional settings we consider a general class of estimators of covariance matrices and precision matrices based on weighted sampling and linear shrinkage the estimation error is measured in terms of both quadratic loss and stein s loss and these loss functions are used to calibrate the set of parameters defining our proposed estimator in an asymptotic setting where the observation dimension is comparable in magnitude to the number of samples we provide estimators of the precision matrix that are as good as their oracle counterparts we test our estimators with both synthetic data and financial market data and monte carlo simulations show the advantage of our precision matrix estimator compared with well known estimators in finite sample size settings

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

shrinkage of covariance matrices for linear signal estimation using cross validation linear estimation of signals is often based on covariance matrices estimated from training which can perform poorly if the training data are limited and the estimated covariance matrices are ill conditioned shrinking the covariance matrix toward a scaled identity matrix can improve the robustness against the model uncertainty provided the shrinkage factor is appropriately chosen this paper introduces several cross validation schemes for choosing the shrinkage factors in applications where the covariance matrices are replaced with sample covariance matrices or constructed from least squares estimates of the linear model parameters for cases where the training and out of training data are identically distributed we derive leave one out cross validation loocv schemes that repeatedly split the training data with respect to time to determine the optimal shrinkage factors for either model fitting or signal estimation for cases where they are distributed differently we develop alternative loocv schemes that repeatedly split the out of training observations with respect to space we derive computationally efficient implementations of those schemes and provide examples to demonstrate their performance

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

analysis of modified smi method for adaptive array weight control the authors characterize the performance of the diagonally loaded sample matrix inverse smi algorithm versus the number k of snapshots used in the covariance matrix estimate by providing o 1 k statistics bias and variance of the array weights output powers and output power ratios such as signal to interference noise ratio sinr and inr the approach accommodates wideband signals monte carlo simulations verify the theoretical analysis

a new approach for mutual information analysis of large dimensional multi antenna channels this paper adresses the behavior of the mutual information of correlated multiple input multiple output mimo rayleigh channels when the numbers of transmit and receive antennas converge to infin at the same rate using a new and simple approach based on poincare nash inequality and on an integration by parts formula it is rigorously established that the mutual information when properly centered and rescaled converges to a standard gaussian random variable simple expressions for the centering and scaling parameters are provided these results confirm previous evaluations based on the powerful but nonrigorous replica method it is believed that the tools that are used in this paper are simple robust and of interest for the communications engineering community

high dimensional covariance matrix estimation using a factor model abstract high dimensionality comparable to sample size is common in many statistical problems we examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases motivated by the arbitrage pricing theory in finance a multi factor model is employed to reduce dimensionality and to estimate the covariance matrix the factors are observable and the number of factors k is allowed to grow with p we investigate the impact of p and k on the performance of the model based covariance matrix estimator under mild assumptions we have established convergence rates and asymptotic normality of the model based estimator its performance is compared with that of the sample covariance matrix we identify situations under which the factor approach increases performance substantially or marginally the impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied the asymptotic results are supported by a thorough simulation study

convergence of the smi and the diagonally loaded smi algorithms with weak interference adaptive array approximations for the power levels at the output of an adaptive array that uses the diagonally loaded sample matrix inversion smi algorithm are derived diagonal loading is a technique where the diagonal of the covariance matrix is augmented with a positive or negative constant prior to inversion the authors examine how the signal to interference plus noise ratio sinr and signal to interference ratio sir at the array output vary with the number of samples taken when the input signals are continuous wave it is shown that positive loading produces more rapid convergence with a reduction in output sir negative loading provides an improved sir level but it is shown that positive loading produces more rapid convergence with a reduction in output sir negative loading provides an improved sir level but it is shown that the output power levels are erratic and slow to converge simulation results which verify the theoretical procedure are given

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

regularized robust estimation of mean and covariance matrix under heavy tailed distributions in this paper the joint mean covariance estimation problem is considered under the scenario that the number of samples is small relative to the problem dimension the samples are assumed drawn independently from a heavy tailed distribution of the elliptical family which can model scenarios where the commonly adopted gaussian assumption is violated either because of the data generating process or the contamination of outliers under the assumption that prior knowledge of the mean and covariance matrix is available we propose a regularized estimator defined as the minimizer of a penalized loss function which combines the prior information and the information provided by the samples the loss function is chosen to be the negative log likelihood function of the cauchy distribution as a conservative representative of heavy tailed distributions and the penalty term is constructed with the prior being its global minimizer the resulting regularized estimator shrinks the mean and the covariance matrix to the prior target the existence and uniqueness of the estimator for finite samples are established under certain regularity conditions numerical algorithms are derived for the estimator based on the majorization minimization framework with guaranteed convergence and simulation results demonstrate that the proposed estimator achieves better estimation accuracy compared to the benchmark estimators

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

performance analysis of beamformers using generalized loading of the covariance matrix in the presence of random steering vector errors robust adaptive beamforming is a key issue in array applications where there exist uncertainties about the steering vector of interest diagonal loading is one of the most popular techniques to improve robustness in this paper we present a theoretical analysis of the signal to interference plus noise ratio sinr for the class of beamformers based on generalized i e not necessarily diagonal loading of the covariance matrix in the presence of random steering vector errors a closed form expression for the sinr is derived that is shown to accurately predict the sinr obtained in simulations this theoretical formula is valid for any loading matrix it provides insights into the influence of the loading matrix and can serve as a helpful guide to select it finally the analysis enables us to predict the level of uncertainties up to which robust beamformers are effective and then depart from the optimal sinr

exact solution of the e e jahn teller and rabi hamiltonian by generalised spheroidal wavefunctions the schrodinger equation for the e e jahn teller and rabi systems in bargmann s hilbert space 1962 is a system of two ordinary differential equations of first order for the spin up and down components of the wavefunctions this system has two regular and one irregular singular points the energy eigenvalues are selected by the requirement that the solutions belong to the space of entire functions the differential equations of the generalised spheroidal wavefunctions have the same singular points and the same exponents at each singular point it is therefore conjectured that the component wavefunctions in the excited state i can be expanded in i 4 generalised spheroidal wavefunctions the energy eigenvalues i i 0 1 5 calculated with the conjectured component wavefunctions agree with numerical values within the computational error the same is true for the coefficients of the neumann expansion of the component wavefunctions a proof is still missing

a generalized potential in the theory of the rabi and e x epsilon jahn teller system the eigenvalue problem for the rabi and the e x epsilon jahn teller hamiltonian in bargmanns hilbert space of analytical functions is a system of two first order differential equations for the two component wavefunctions whose entire solutions the eigenstates are sought we show that each eigenstate is a terminating series in the derivatives of a scalar entire function d z called the generalized potential which satisfies a higher order differential equation the coefficients of the terminating series depend on the physical parameters and are polynomials in the independent variable z the coefficients are identical in all eigenstates

finite sample linear filter optimization in wireless communications and financial systems we study the problem of linear filter optimization with finite sample size which has wide applications such as beamformer design in wireless communications and portfolio optimization in finance traditional methods in both fields are not robust against the imprecise channel vector and the noise covariance matrix or the mean return and the covariance of assets in finance due to finite sample size we consider estimation errors both in the channel vector and the noise covariance matrix or the mean return and the covariance simultaneously we resort to high dimensional asymptotics to account for the fact that the observation dimension is of the same order of magnitude as the number of samples and use the diagonal loading method or the shrinkage estimator to improve the robustness the channel vector or mean return and the noise covariance matrix are estimated from the training data and then corrected under several widely used criteria in an asymptotic setting where the number of samples is comparable to the observation dimension we obtain linear filters that are as good as the optimal filters with a shrinkage structure and a perfect channel vector or mean return under different criteria monte carlo simulations show the advantage of our linear filters in the finite sample size regime

a generalized potential in the theory of the rabi and e x epsilon jahn teller systems ii for pt i see reik ibid vol 26 p 6549 1993 the eigenvalue problem for the rabi and e x epsilon jahn teller hamiltonians in bargmann s hilbert space is a system of two first order differential equations for the two component wavefunctions for which entire solutions are sought the concept of the generalized potential has been introduced in a previous paper together with a particular example here we treat a simpler potential n z which satisfies a second order ordinary differential equation closely related to the differential equation of the confluent heun functions the component wavefunctions are linear in the potential and its first derivative the coefficients of n z and dn z dz are functions of z and the physical parameters which are identical in all eigenstates the relation to the previous example is fully discussed

the non commutative specker phenomenon in the uncountable case isthecanonicalprojection previously the second author showed the failure of the specker phe nomenon in the uncountable case in a di erent situation 6 see also 1 we explain the di erence between this result and theorem 1 1 ofthe present paper there is a canonical subgroup of the unrestrictedfree product which is called the free complete product and denoted by

optimal rates of convergence for covariance matrix estimation covariance matrix plays a central role in multivariate statistical analysis significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices however a minimax theory has yet been developed in this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and frobenius norm it is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation the minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties a key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound the technical analysis requires new ideas that are quite different from those used in the more conventional function sequence estimation problems

estimation of a covariance matrix under stein s loss on decrit la technique generale de stein pour ameliorer l estimateur sans biais invariant le meilleur et l estimateur minimax de la matrice de covariance normale

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

generalized consistent error estimator of linear discriminant analysis a classifier is epistemologically vacuous without an accurate estimate of its true error rate in situations where the number of sample points is of the same order of magnitude as the dimension of observations serious issues arise with respect to the performance of error estimators in this paper we place the problem of synthesizing an error rate estimator of a common linear classifier in an asymptotic setting in which the number of sample points is kept comparable in magnitude to the dimension of observations double asymptotic we construct a generalized consistent estimator of the true error rate for linear discriminant analysis in the multivariate gaussian model under the assumption of a common covariance matrix in other words the estimator converges to true error rate in the double asymptotic sense we employ simulations using both synthetic and real data to compare the performance of the new estimator to the classical consistent estimator of the true error plug in estimator as well as other well known estimators we observe that the constructed estimator can outperform other estimators of the true error in many situations in terms of bias and root mean square rms error

quasi exact solution of the rabi hamiltonian a method is suggested to obtain the quasi exact solution of the rabi hamiltonian it is conceptually simple and can be easily extended to other systems the analytical expressions are obtained for eigenstates and eigenvalues in terms of orthogonal polynomials it is also demonstrated that the rabi system in a particular case coincides with the quasi exactly solvable poschl teller potential

robust adaptive beamforming for general rank signal models the performance of adaptive beamforming methods is known to degrade severely in the presence of even small mismatches between the actual and presumed array responses to the desired signal such mismatches may frequently occur in practical situations because of violation of underlying assumptions on the environment sources or sensor array this is especially true when the desired signal components are present in the beamformer training data snapshots because in this case the adaptive array performance is very sensitive to array and model imperfections the similar phenomenon of performance degradation can occur even when the array response to the desired signal is known exactly but the training sample size is small we propose a new powerful approach to robust adaptive beamforming in the presence of unknown arbitrary type mismatches of the desired signal array response our approach is developed for the most general case of an arbitrary dimension of the desired signal subspace and is applicable to both the rank one point source and higher rank scattered source fluctuating wavefront desired signal models the proposed robust adaptive beamformers are based on explicit modeling of uncertainties in the desired signal array response and data covariance matrix as well as worst case performance optimization simple closed form solutions to the considered robust adaptive beamforming problems are derived our new beamformers have a computational complexity comparable with that of the traditional adaptive beamforming algorithms while at the same time offer a significantly improved robustness and faster convergence rates

performance analysis of direction finding with large arrays and finite data this paper considers analysis of methods for estimating the parameters of narrow band signals arriving at an array of sensors this problem has important applications in for instance radar direction finding and underwater source localization the so called deterministic and stochastic maximum likelihood ml methods are the main focus of this paper a performance analysis is carried out assuming a finite number of samples and that the array is composed of a sufficiently large number of sensors several thousands of antennas are not uncommon in e g radar applications strong consistency of the parameter estimates is proved and the asymptotic covariance matrix of the estimation error is derived unlike the previously studied large sample case the present analysis shows that the accuracy is the same for the two ml methods furthermore the asymptotic covariance matrix of the estimation error coincides with the deterministic cramer rao bound under a certain assumption the ml methods can be implemented by means of conventional beamforming for a large enough number of sensors we also include a simple simulation study which indicates that both ml methods provide efficient estimates for very moderate array sizes whereas the beamforming method requires a somewhat larger array aperture to overcome the inherent bias and resolution problem

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

finite sample size effect on minimum variance beamformers optimum diagonal loading factor for large arrays minimum variance beamformers are usually complemented with diagonal loading techniques in order to provide robustness against several impairments such as imprecise knowledge of the steering vector or finite sample size effects this paper concentrates on this last application of diagonal loading techniques i e it is assumed that the steering vector is perfectly known and that diagonal loading is used to alleviate the finite sample size impairments the analysis herein is asymptotic in the sense that it is assumed that both the number of antennas and the number of samples are high but have the same order of magnitude borrowing some results of random matrix theory the authors first derive a deterministic expression that describes the asymptotic signal to noise plus interference ratio sinr at the output of the diagonally loaded beamformer then making use of the statistical theory of large observations also known as general statistical analysis or g analysis the authors derive an estimator of the optimum loading factor that is consistent when both the number of antennas and the sample size increase without bound at the same rate because of that the estimator has an excellent performance even in situations where the quotient between the number of observations is low relative to the number of elements of the array

using spectral estimation techniques in adaptive processing antenna systems improved spectral estimation techniques hold promise for becoming a valuable asset in adaptive processing array antenna systems their value lies in the considerable amount of additional useful information which they can provide about the interference environment utilizing a relatively small number of degrees of freedom dof the superresolution capabilities estimation of coherence and relative power level determination serve to complement and refine the data from faster conventional estimation techniques two conceptual application area examples for using such techniques are discussed partially adaptive low sidelobe arrays and fully adaptive tracking arrays for the partially adaptive area the information is utilized for efficient assignment of a limited number of dof in a beamspace constrained adaptive system in order to obtain a stable main beam retention of low sidelobes considerably faster response and reduction in overall cost these benefits are demonstrated via simulation examples computed for a 16 element linear array for the fully adaptive tracking array area the information is utilized in an all digital processing system concept to permit stable hulling of coherent interference sources in the main beam region efficient assignment control of the available dof and greater flexibility in time domain adaptive filtering strategy

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

exact solution of non adiabatic model hamiltonians in solid state physics and optics simple non adiabatic model hamiltonians are treated using the realisation of the vibrational modes on bargmann s hilbert space of analytic functions in this formulation the schrodinger equation is a system of linear first order differential equations the energy eigenvalues are selected by the requirement that the solutions belong to the space of entire functions the solutions are given in terms of neumann series the recurrence relations for the expansion coefficients have a simple structure under particular conditions for the interaction constant they allow for terminating neumann series isolated exact solutions in the general case the conditions for the eigenvalues are transcendental equations involving a continued fraction the continued fraction can be approximated to any desired degree of accuracy in a rapidly convergent process based on worpitzky s theorem and its relation to conformal mapping the eigenvalues are calculated a physical interpretation of the solutions which makes use of intuitive arguments is also given

homomorphisms on infinite direct product algebras especially lie algebras abstract we study surjective homomorphisms f i a i b of not necessarily associative algebras over a commutative ring k for i a generally infinite set especially when k is a field and b is countable dimensional over k our results have the following consequences when k is an infinite field the algebras are lie algebras and b is finite dimensional if all the lie algebras a i are solvable then so is b if all the lie algebras a i are nilpotent then so is b if k is not of characteristic 2 or 3 and all the lie algebras a i are finite dimensional and are direct products of simple algebras then i so is b ii f splits and iii under a weak cardinality bound on i f is continuous in the pro discrete topology a key fact used in getting i iii is that over any such field every finite dimensional simple lie algebra l can be written l x 1 l x 2 l for some x 1 x 2 l which we prove from a recent result of j m bois the general technique of the paper involves studying conditions under which a homomorphism on i a i must factor through the direct product of finitely many ultraproducts of the a i several examples are given and open questions noted