fast estimators of time delay and doppler stretch based on discrete time methods estimation of the time delay and the doppler stretch of a signal is required in several signal processing applications this paper is focused on the joint fine estimation of these two parameters by a fast interpolation of the estimated ambiguity function four discrete time methods viz multirate piecewise scaling linear scaling and indirect estimators based on an orthogonal model are introduced their mean square error is mathematically derived for random signals corrupted by additive random noises the obtained expressions have been evaluated for some typical parameter sets in the reference case of gaussian signals and noises the numerical results compared with the accuracy of a continuous time estimator show the near efficiency of the multirate estimator for a wide range of snrs in fact the adjustable multirate estimator can operate under near optimal conditions unlike the approximate i e piecewise and linear scaling and the indirect methods based on constant sampling rates

distributionally robust optimization and its tractable approximations in this paper we focus on a linear optimization problem with uncertainties having expectations in the objective and in the set of constraints we present a modular framework to obtain an approximate solution to the problem that is distributionally robust and more flexible than the standard technique of using linear rules our framework begins by first affinely extending the set of primitive uncertainties to generate new linear decision rules of larger dimensions and is therefore more flexible next we develop new piecewise linear decision rules that allow a more flexible reformulation of the original problem the reformulated problem will generally contain terms with expectations on the positive parts of the recourse variables finally we convert the uncertain linear program into a deterministic convex program by constructing distributionally robust bounds on these expectations these bounds are constructed by first using different pieces of information on the distribution of the underlying uncertainties to develop separate bounds and next integrating them into a combined bound that is better than each of the individual bounds

scalable estimation and regularization for the logistic normal multinomial model clustered multinomial data are prevalent in a variety of applications such as microbiome studies where metagenomic sequencing data are summarized as multinomial counts for a large number of bacterial taxa per subject count normalization with ad hoc zero adjustment tends to result in poor estimates of abundances for taxa with zero or small counts to account for heterogeneity and overdispersion in such data we suggest using the logistic normal multinomial lnm model with an arbitrary correlation structure to simultaneously estimate the taxa compositions by borrowing information across subjects we overcome the computational difficulties in high dimensions by developing a stochastic approximation em algorithm with hamiltonian monte carlo sampling for scalable parameter estimation in the lnm model the ill conditioning problem due to unstructured covariance is further mitigated by a covariance regularized estimator with a condition number constraint the advantages of the proposed methods are illustrated through simulations and an application to human gut microbiome data

nonparametric eigenvalue regularized precision or covariance matrix estimator we introduce nonparametric regularization of the eigenvalues of a sample covariance matrix through splitting of the data nercome and prove that nercome enjoys asymptotic optimal nonlinear shrinkage of eigenvalues with respect to the frobenius norm one advantage of nercome is its computational speed when the dimension is not too large we prove that nercome is positive definite almost surely as long as the true covariance matrix is so even when the dimension is larger than the sample size with respect to the stein s loss function the inverse of our estimator is asymptotically the optimal precision matrix estimator asymptotic efficiency loss is defined through comparison with an ideal estimator which assumed the knowledge of the true covariance matrix we show that the asymptotic efficiency loss of nercome is almost surely 0 with a suitable split location of the data we also show that all the aforementioned optimality holds for data with a factor structure our method avoids the need to first estimate any unknowns from a factor model and directly gives the covariance or precision matrix estimator which can be useful when factor analysis is not the ultimate goal we compare the performance of our estimators with other methods through extensive simulations and real data analysis

on reduced semidefinite programs for second order moment bounds with applications we show that the complexity of computing the second order moment bound on the expected optimal value of a mixed integer linear program with a random objective coefficient vector is closely related to the complexity of characterizing the convex hull of the points 1 atopwithdelims varvec x 1 atopwithdelims varvec x varvec x in mathcal x 1x1x x x where mathcal x x is the feasible region in fact we can replace the completely positive programming formulation for the moment bound on mathcal x x with an associated semidefinite program provided we have a linear or a semidefinite representation of this convex hull as an application of the result we identify a new polynomial time solvable semidefinite relaxation of the distributionally robust multi item newsvendor problem by exploiting results from the boolean quadric polytope for mathcal x x described explicitly by a finite set of points our formulation leads to a reduction in the size of the semidefinite program we illustrate the usefulness of the reduced semidefinite programming bounds in estimating the expected range of random variables with two applications arising in random walks and best worst choice models

distributionally robust convex optimization distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty the distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker s prior information in this paper we propose a unifying framework for modeling and solving distributionally robust optimization problems we introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold these ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases they also allow us to characterize distributional families in terms of several classical and or robust statistical indicators that have not yet been studied in the context of robust optimization we determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable we also provide tractable conservative approximations for problems that violate these conditions

estimation of a covariance matrix using the reference prior estimation of a covariance matrix sum is a notoriously difficult problem the standard unbiased estimator can be substantially suboptimal we approach the problem from a noninformative prior bayesian perspective developing the reference noninformative prior for a covariance matrix and obtaining expressions for the resulting bayes estimators these expressions involve the computation of high dimensional posterior expectations which is done using a recent markov chain simulation tool the hit and run sampler frequentist risk comparisons with previously suggested estimators are also given and determination of the accuracy of the estimators is addressed

time delay estimation by generalized cross correlation methods the problem of estimating time delay by cross correlation methods is reexamined for the whole class of stationary signals expressions are derived for the estimation mean square error mse by the cross correlation method and are shown to be identical to previously published results for gaussian signals the generalized cross correlation method is also analyzed and the optimal weight function for this method is derived it is shown to be identical to that derived for gaussian signals by the maximum likelihood method for the cross correlation method a simplified mse expression is derived which is to be used instead of a previously published result

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

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

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

subspace based estimation of time delays and doppler shifts this paper considers the problem of estimating the time delays and doppler shifts of a known waveform received via several distinct paths by an array of antennas the general maximum likelihood estimator is presented and is shown to require a 2d dimensional nonlinear minimization where d is the number of received signal reflections two alternative solutions based on signal and noise subspace fitting are proposed requiring only a d dimensional minimization in particular we show how to decouple the required search into a two step procedure where the delays are estimated and the dopplers solved for explicitly initial conditions for the time delay search can be obtained by applying generalizations of the music and esprit algorithms which are also outlined in the paper simulation examples are included to illustrate the algorithms performance relative to the cramer rao bound

a convex pseudolikelihood framework for high dimensional partial correlation estimation with convergence guarantees author s khare k oh sy rajaratnam b abstract 2014 royal statistical society sparse high dimensional graphical model selection is a topic of much interest in modern day statistics a popular approach is to apply l1 penalties to either parametric likelihoods or regularized regression pseudolikelihoods with the latter having the distinct advantage that they do not explicitly assume gaussianity as none of the popular methods proposed for solving pseudolikelihood based objective functions have provable convergence guarantees it is not clear whether corresponding estimators exist or are even computable or if they actually yield correct partial correlation graphs we propose a new pseudolikelihood based graphical model selection method that aims to overcome some of the shortcomings of current methods but at the same time retain all their respective strengths in particular we introduce a novel framework that leads to a convex formulation of the partial covariance regression graph problem resulting in an objective function comprised of quadratic forms the objective is then optimized via a co ordinatewise approach the specific functional form of the objective function facilitates rigorous convergence analysis leading to convergence guarantees an important property that cannot be established by using standard results when the dimension is larger than the sample size as is often the case in high dimensional applications these convergence guarantees ensure that estimators are well defined under very general conditions and are always computable in addition the approach yields estimators that have good large sample properties and also respect symmetry furthermore application to simulated and real data timing comparisons and numerical convergence is demonstrated we also present a novel unifying framework that places all graphical pseudolikelihood methods as special cases of a more general formulation leading to important insights

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

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

computing bounds for stochastic programming problems by means of a generalized moment problem bounds on the expected value of a convex function are obtained by means of an approximating generalized moment problem numerical implementation is discussed in the context of stochastic programming problems

distributionally robust optimization with polynomial densities theory models and algorithms in distributionally robust optimization the probability distribution of the uncertain problem parameters is itself uncertain and a fictitious adversary e g nature chooses the worst distribution from within a known ambiguity set a common shortcoming of most existing distributionally robust optimization models is that their ambiguity sets contain pathological discrete distributions that give nature too much freedom to inflict damage we thus introduce a new class of ambiguity sets that contain only distributions with sum of squares sos polynomial density functions of known degrees we show that these ambiguity sets are highly expressive as they conveniently accommodate distributional information about higher order moments conditional probabilities conditional moments or marginal distributions exploiting the theoretical properties of a measure based hierarchy for polynomial optimization due to lasserre siam j optim 21 3 864 885 2011 we prove that certain worst case expectation constraints are polynomial time solvable under these new ambiguity sets we also show how sos densities can be used to approximately solve the general problem of moments we showcase the applicability of the proposed approach in the context of a stylized portfolio optimization problem and a risk aggregation problem of an insurance company

joint tdoa and fdoa estimation a conditional bound and its use for optimally weighted localization modern passive emitter location systems are often based on joint estimation of the time difference of arrival tdoa and frequency difference of arrival fdoa of an unknown signal at two or more sensors classical derivation of the associated cramer rao bound crb relies on a stochastic stationary gaussian signal model leading to a diagonal fisher information matrix with respect to the tdoa and fdoa this diagonality implies that under asymptotic conditions the respective estimation errors are uncorrelated however for some specific nonstationary non gaussian signals especially chirp like signals these errors can be strongly correlated in this work we derive a conditional or a signal specific crb modeling the signal as a deterministic unknown given any particular signal our crb reflects the possible signal induced correlation between the tdoa and fdoa estimates in addition to its theoretical value we show that the resulting crb can be used for optimal weighting of tdoa fdoa pairs estimated over different signal intervals when combined for estimating the target location substantial improvement in the resulting localization accuracy is shown to be attainable by such weighting in a simulated operational scenario with some chirp like target signals

sparse estimation of a covariance matrix we suggest a method for estimating a covariance matrix on the basis of a sample of vectors drawn from a multivariate normal distribution in particular we penalize the likelihood with a lasso penalty on the entries of the covariance matrix this penalty plays two important roles it reduces the effective number of parameters which is important even when the dimension of the vectors is smaller than the sample size since the number of parameters grows quadratically in the number of variables and it produces an estimate which is sparse in contrast to sparse inverse covariance estimation our method s close relative the sparsity attained here is in the covariance matrix itself rather than in the inverse matrix zeros in the covariance matrix correspond to marginal independencies thus our method performs model selection while providing a positive definite estimate of the covariance the proposed penalized maximum likelihood problem is not convex so we use a majorize minimize approach in which we iteratively solve convex approximations to the original nonconvex problem we discuss tuning parameter selection and demonstrate on a flow cytometry dataset how our method produces an interpretable graphical display of the relationship between variables we perform simulations that suggest that simple elementwise thresholding of the empirical covariance matrix is competitive with our method for identifying the sparsity structure additionally we show how our method can be used to solve a previously studied special case in which a desired sparsity pattern is prespecified

uncertain convex programs randomized solutions and confidence levels many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or instance parameter two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty robust optimization and chance constrained optimization robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances in chance constrained optimization a probability distribution is instead assumed on the uncertain parameters and the constraints are enforced up to a pre specified level of probability unfortunately however both approaches lead to computationally intractable problem formulations

design free estimation of variance matrices abstract this paper introduces a new method for estimating variance matrices starting from the orthogonal decomposition of the sample variance matrix we exploit the fact that orthogonal matrices are never ill conditioned and therefore focus on improving the estimation of the eigenvalues we estimate the eigenvectors from just a fraction of the data then use them to transform the data into approximately orthogonal series that deliver a well conditioned estimator by construction even when there are fewer observations than dimensions we also show that our estimator has lower error norms than the traditional one our estimator is design free we make no assumptions on the distribution of the random sample or on any parametric structure the variance matrix may have simulations confirm our theoretical results and they also show that our simple estimator does very well in comparison with other existing methods

on the cramer rao bound for time delay and doppler estimation corresp using a theorem due to whittle simple derivations of the cramer rao lower bound are presented for some delay estimation problems related to a single source multiple sources and multipath the problem of doppler estimation is briefly discussed

on the rate of convergence of optimal solutions of monte carlo approximations of stochastic programs in this paper we discuss monte carlo simulation based approximations of a stochastic programming problem we show that if the corresponding random functions are convex piecewise linear and the distribution is discrete then an optimal solution of the approximating problem provides an exact optimal solution of the true problem with probability one for sufficiently large sample size moreover by using the theory of large deviations we show that the probability of such an event approaches one exponentially fast with increase of the sample size in particular this happens in the case of linear two or multi stage stochastic programming with recourse if the corresponding distributions are discrete the obtained results suggest that in such cases monte carlo simulation based methods could be very efficient we present some numerical examples to illustrate the ideas involved

on constraint sampling in the linear programming approach to approximate dynamic programming in the linear programming approach to approximate dynamic programming one tries to solve a certain linear program the alp that has a relatively small numberk of variables but an intractable numberm of constraints in this paper we study a scheme that samples and imposes a subset ofm

exact algorithms for the chance constrained vehicle routing problem we study the chance constrained vehicle routing problem ccvrp a version of the vehicle routing problem vrp with stochastic demands where a limit is imposed on the probability that each vehicle s capacity is exceeded a distinguishing feature of our proposed methodologies is that they allow correlation between random demands whereas nearly all existing exact methods for the vrp with stochastic demands require independent demands we first study an edge based formulation for the ccvrp in particular addressing the challenge of how to determine a lower bound on the number of vehicles required to serve a subset of customers we then investigate the use of a branch and cut and price bcp algorithm while bcp algorithms have been considered the state of the art in solving the deterministic vrp few attempts have been made to extend this framework to the vrp with stochastic demands in contrast to the deterministic vrp we find that the pricing problem for the ccvrp problem is strongly mathcal np hard even when the routes being priced are allowed to have cycles we therefore propose a further relaxation of the routes that enables pricing via dynamic programming we also demonstrate how our proposed methodologies can be adapted to solve a distributionally robust ccvrp problem numerical results indicate that the proposed methods can solve instances of ccvrp having up to 55 vertices

algorithms for ambiguity function processing calculation of the complex ambiguity function is viewed as the basis for joint estimation of the differential delay and differential frequency offset between two waveforms that contain a common component plus additive noise in many applications the required accuracy leads to a need for integration over long data sets that can become a challenge for near real time digital processing the nature of the ambiguity processing is interpreted and an algorithm approach is shown that minimizes the processing burden over a broad category of applications without affecting performance

the joint estimation of differential delay doppler and phase corresp in radio and sonar applications it sometimes happens that narrow band signals originated from a remote source and observed at a pair of receivers differ by unknown differential phase and doppler shift in addition to the differential delay corresponding to the range difference the correspondence presents the joint maximum likelihood ml estimate of the differential delay doppler and phase and examines their accuracy by deriving the cramer rao bound it is shown that the joint ml estimators are the values of the delay and doppler that maximize the magnitude of a generalized ambiguity function analogous to the one used in radar it is also shown that for long observation time and high enough signal to noise ratio there is no degradation in the accuracy of the time delay estimator due to the additional phase and doppler uncertainty and that the differential doppler is uncorrelated with the differential delay and phase estimators

robust optimization of sums of piecewise linear functions with application to inventory problems robust optimization is a methodology that has gained a lot of attention in the recent years this is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models unfortunately the second property is usually lost when the cost function that needs to be robustified is not concave or linear with respect to the perturbing parameters in this paper we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set given that these problems are known to be intractable we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed integer linear program and relate this scheme to methods that are based on exploiting affine decision rules our new scheme gives rise to two tractable models that respectively take the shape of a linear program and a semidefinite program with the latter having the potential to provide solutions of better quality than the former at the price of heavie

group centre and group median of a tree abstract we consider the problem of identifying group centre and group median of a tree algorithms are presented that are better than the available algorithms which solve these problems on general networks for the group median problem an alternate approach is suggested which runs in linear time on specially structured problems properties of the set of all optimal solutions of these problems are also investigated

stochastic optimization problems with incomplete information on distribution functions the main purpose of this paper is to discuss numerical optimization procedures based on duality theory for stochastic extremal problems in which the distribution function is only partially known we formulate such problems as minimax problems in which the inner problem involves optimization with respect to probability measures the latter problem is solved using generalized linear programming techniques then we state the dual problem to the initial stochastic optimization problem numerical procedures that avoid the difficulties associated with solving the inner problem are proposed

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

comment ridge regression and regularization of large matrices we view ridge regression through the lens of eigenvalue shrinkage and consider its influence on two modern problems in high dimensional statistical inference covariance estimation and community d

worst case distribution analysis of stochastic programs we show that for even quasi concave objective functions the worst case distribution with respect to a family of unimodal distributions of a stochastic programming problem is a uniform distribution this extends the so called uniformity principle of barmish and lagoa 1997 where the objective function is the indicator function of a convex symmetric set

optimization with reference based robust preference constraints we propose an optimization model using a novel robust preference relationship reference based almost stochastic dominance rsd the concept of rsd addresses the two problems in utility based dec

improvement of tdoa position fixing using the likelihood curvature a conventional multilateration is a two step process where in the first step the time difference of arrivals tdoa of a signal at multiple sensors are estimated and in the second step these tdoas are used in some position fixing technique to estimate the location of an emitter several estimators have been proposed over the years for the estimation of the tdoas many techniques have been proposed for position fixing as well much of the research on position fixing has been focused on obtaining a simplified closed form solution for the unknown deterministic signal model stein had derived the maximum likelihood estimator mle for the tdoa between two sensors which is the peak location of the cross correlation function since the asymptotic variance of an mle approaches the cramer rao lower bound crlb which is the inverse of the negative of the expected value of the curvature of the log likelihood function using this as a motivation we propose a weighted least squares type position fixing technique where the weights are computed from the curvature of the log likelihood function

wishart distributions for decomposable covariance graph models gaussian covariance graph models encode marginal independence among the components of a multivariate random vector by means of a graph g these models are distinctly different from the traditional concentration graph models often also referred to as gaussian graphical models or covariance selection models since the zeros in the parameter are now reflected in the covariance matrix as compared to the concentration matrix 1 the parameter space of interest for covariance graph models is the cone p g of positive definite matrices with fixed zeros corresponding to the missing edges of g as in letac and massam ann statist 35 2007 1278 1323 we consider the case where g is decomposable in this paper we construct on the cone p g a family of wishart distributions which serve a similar purpose in the covariance graph setting as those constructed by letac and massam ann statist 35 2007 1278 1323 and dawid and lauritzen ann statist 21 1993 1272 1317 do in the concentration graph setting we proceed to undertake a rigorous study of these covariance wishart distributions and derive several deep and useful properties of this class first they form a rich conjugate family of priors with multiple shape parameters for covariance graph models second we show how to sample from these distributions by using a block gibbs sampling algorithm and prove convergence of this block gibbs sampler development of this class of distributions enables bayesian inference which in turn allows for the estimation of e even in the case when the sample size is less than the dimension of the data i e when n p otherwise not generally possible in the maximum likelihood framework third we prove that when g is a homogeneous graph our covariance priors correspond to standard conjugate priors for appropriate directed acyclic graph dag models this correspondence enables closed form expressions for normalizing constants and expected values and also establishes hyper markov properties for our class of priors we also note that when g is homogeneous the family iw qg of letac and massam ann statist 35 2007 1278 1323 is a special case of our covariance wishart distributions fourth and finally we illustrate the use of our family of conjugate priors on real and simulated data

robust portfolio selection problems in this paper we show how to formulate and solve robust portfolio selection problems the objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters we introduce uncertainty structures for the market parameters and show that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second order cone programs and therefore the computational effort required to solve them is comparable to that required for solving convex quadratic programs moreover we show that these uncertainty structures correspond to confidence regions associated with the statistical procedures employed to estimate the market parameters finally we demonstrate a simple recipe for efficiently computing robust portfolios given raw market data and a desired level of confidence

maximum likelihood estimation of time varying delay part i this paper presents for the first time the exact theoretical solution to the problem of maximum likelihood ml estimation of time varying delay d t between a random signal s t received at one point in the presence of uncorrelated noise and the time delayed scaled version s t d t of that signal received at another point in the presence of uncorrelated noise the signal is modeled as a sample function of a nonstationary gaussian random process and the observation interval is arbitrary the analysis of this paper represents a generalization of that of knapp and carter 1 who derived the ml estimator for the case that the delay is constant d t d 0 the signal process is stationary and the received processes are observed over the infinite interval we show that the ml estimator of d t can be implemented in any of four canonical forms which in general are time varying systems we also show that our results reduce to a generalized cross correlator for the special case treated in 1

complexity analysis of the analytic center cutting plane method that uses multiple cuts we analyze the complexity of the analytic center cutting plane or column generation algorithm for solving general convex problems defined by a separation oracle the oracle is called at the analytic center of a polytope which contains a solution set and is given by the intersection of the linear inequalities previously generated from the oracle if the center is not in the solution set separating hyperplanes will be placed through the center to shrink the containing polytope while the complexity result has been recently established for the algorithm when one cutting plane is placed in each iteration the result remains open when multiple cuts are added moreover adding multiple cuts actually is a key to practical effectiveness in solving many problems and it presents theoretical difficulties in analyzing cutting plane methods in this paper we show that the analytic center cutting plane algorithm with multiple cuts added in each iteration still is a fully polynomial approximation algorithm

nonlinear shrinkage estimation of large dimensional covariance matrices many statistical applications require an estimate of a covariance matrix and or its inverse when the matrix dimension is large compared to the sample size which happens frequently the sample covariance matrix is known to perform poorly and may suffer from ill conditioning there already exists an extensive literature concerning improved estimators in such situations in the absence of further knowledge about the structure of the true covariance matrix the most successful approach so far arguably has been shrinkage estimation shrinking the sample covariance matrix to a multiple of the identity by taking a weighted average of the two turns out to be equivalent to linearly shrinking the sample eigenvalues to their grand mean while retaining the sample eigenvectors our paper extends this approach by considering nonlinear transformations of the sample eigenvalues we show how to construct an estimator that is asymptotically equivalent to an oracle estimator suggested in previous work as demonstrated in extensive monte carlo simulations the resulting bona fide estimator can result in sizeable improvements over the sample covariance matrix and also over linear shrinkage

efficient algorithms for solving condition number constrained matrix minimization problems abstract well conditioned matrices are often required in science and engineering such as signal processing and finance problem to find the nearest positive definite matrix by explicitly imposing a constraint on the condition number are considered in this paper a new algorithm based on geometric perspective is proposed for getting the required well conditioned matrix based on these a condition number constrained matrix minimization problem is further considered where the constraints are imposed for avoiding degenerate solutions in which parameter matrices become rank deficient an inexact version of alternating direction method with truly implementable inexactness criteria is proposed for solving this problem numerical experiments including comparison with some existing methods are performed to illustrate the efficiency of the proposed algorithms

proper quaternion gaussian graphical models in this paper we extend gaussian graphical models to proper quaternion gaussian distributions the properness assumption reduces the number of unknowns by a factor of four while graphical models reduce the number of degrees of freedom via sparsity each of the methods allows accurate estimation using a small number of samples to enjoy both gains we show that the proper quaternion gaussian inverse covariance estimation problem is convex and has a closed form solution we proceed to demonstrate that the additional sparsity constraints on the inverse covariance matrix also lead to a convex problem and the optimizations can be efficiently solved by standard numerical methods in the special but practical case of a chordal graph we provide a closed form solution we demonstrate the improved performance of our suggested estimators on both synthetic and real data

shrinkage estimators for covariance matrices estimation of covariance matrices in small samples has been studied by many authors standard estimators like the unstructured maximum likelihood estimator ml or restricted maximum likelihood reml estimator can be very unstable with the smallest estimated eigenvalues being too small and the largest too big a standard approach to more stably estimating the matrix in small samples is to compute the ml or reml estimator under some simple structure that involves estimation of fewer parameters such as compound symmetry or independence however these estimators will not be consistent unless the hypothesized structure is correct if interest focuses on estimation of regression coefficients with correlated or longitudinal data a sandwich estimator of the covariance matrix may be used to provide standard errors for the estimated coefficients that are robust in the sense that they remain consistent under misspecification of the covariance structure with large matrices however the inefficiency of the sandwich estimator becomes worrisome we consider here two general shrinkage approaches to estimating the covariance matrix and regression coefficients the first involves shrinking the eigenvalues of the unstructured ml or reml estimator the second involves shrinking an unstructured estimator toward a structured estimator for both cases the data determine the amount of shrinkage these estimators are consistent and give consistent and asymptotically efficient estimates for regression coefficients simulations show the improved operating characteristics of the shrinkage estimators of the covariance matrix and the regression coefficients in finite samples the final estimator chosen includes a combination of both shrinkage approaches i e shrinking the eigenvalues and then shrinking toward structure we illustrate our approach on a sleep eeg study that requires estimation of a 24 x 24 covariance matrix and for which inferences on mean parameters critically depend on the covariance estimator chosen we recommend making inference using a particular shrinkage estimator that provides a reasonable compromise between structured and unstructured estimators

robust and stochastically weighted multiobjective optimization models and reformulations we introduce and study a family of models for multiexpert multiobjective criteria decision making these models use a concept of weight robustness to generate a risk averse decision in particular the multiexpert multicriteria robust weighted sum approach mcrow introduced in this paper identifies a robust pareto decision that minimizes the worst case weighted sum of objectives over a given weight region the corresponding objective value called the robust value of a decision is shown to be increasing and concave in the weight set we study compact reformulations of the mcrow model with polyhedral and conic descriptions of the weight regions the mcrow model is developed further for stochastic multiexpert multicriteria decision making by allowing ambiguity or randomness in the weight region as well as the objective functions the properties of the proposed approach are illustrated with a few textbook examples the usefulness of the stochastic mcrow model is demonstrated using a disaster planning example and an agriculture revenue management example

solving convex programs by random walks minimizing a convex function over a convex set in n dimensional space is a basic general problem with many interesting special cases here we present a simple new algorithm for convex optimization based on sampling by a random walk it extends naturally to minimizing quasi convex functions and to other generalizations

penalized normal likelihood and ridge regularization of correlation and covariance matrices high dimensionality causes problems in various areas of statistics a particular situation that rarely has been considered is the testing of hypotheses about multivariate regression models in which

differential delay doppler ml estimation with unknown signals previously published analyses on the maximum likelihood estimation of joint frequency and time offsets between two noisy versions of a common signal have assumed a gaussian random signal with a known power spectrum in many applications the common signal is not gaussian and there may be no prior knowledge of its detailed structure instead estimation of a hypothesized common signal can be construed as another element in the estimation process the analysis is straightforward and shows that the complex ambiguity function caf still represents the ml approach when the noise is gaussian and spectrally flat additional interpretation shows that use of interference rejection filtering followed by a caf is an attractive suboptimum approach in an environment of narrowband interferers

sparse gaussian graphical model estimation via alternating minimization summaryseveral methods have recently been proposed for estimating sparse gaussian graphical models using ell 1 regularization on the inverse covariance or precision matrix despite recent advances contemporary applications require even faster methods to handle ill conditioned high dimensional datasets in this paper we propose a new method for solving the sparse inverse covariance estimation problem using the alternating minimization algorithm which effectively works as a proximal gradient algorithm on the dual problem our approach has several advantages it is faster than state of the art algorithms by many orders of magnitude its global linear convergence has been rigorously demonstrated underscoring its good theoretical properties it facilitates additional constraints on pairwise or marginal relationships between feature pairs based on domain specific knowledge and it is better at handling extremely ill conditioned problems our algorithm is shown to be more accurate and faster on simulated and real datasets

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

optimal inequalities in probability theory a convex optimization approach we propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for p x in s for a set s defined by polynomial inequalities and a random vector x defined on omega subseteq cal r n that has a given collection of up to kth order moments in the univariate case we provide optimal bounds on p x in s when the first k moments of x are given as the solution of a semidefinite optimization problem in k 1 dimensions in the multivariate case if the sets s and omega are given by polynomial inequalities we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n for fixed k we characterize the complexity of the problem of deriving tight moment inequalities we show that it is np hard to find tight bounds for k geq 4 and omega cal r n and for k geq 2 and omega cal r n when the data in the problem is rational for k 1 and omega cal r n we show that we can find tight upper bounds by solving n convex optimization problems when the set s is convex and we provide a polynomial time algorithm when s and omega are unions of convex sets over which linear functions can be optimized efficiently for the case k 2 and omega cal r n we present an efficient algorithm for finding tight bounds when s is a union of convex sets over which convex quadratic functions can be optimized efficiently

group centre and group median of a network abstract the absolute centre and median of a network can be computed in polynomial time we use a partition of the nodes to define group centre and group median of the network these ideas can be used to unify the concepts of absolute centre and median we give polynomially bounded algorithms to identify group centre and group median of a network it may be observed that the introduction of partitions in some other combinatorial optimization problems e g the assignment problem changes the polynomial nature of the problem

solving semidefinite quadratic linear programs using sdpt3 this paper discusses computational experiments with linear optimization problems involving semidefinite quadratic and linear cone constraints sqlps many test problems of this type are solved using a new release of sdpt3 a matlab implementation of infeasible primal dual path following algorithms the software developed by the authors uses mehrotra type predictor corrector variants of interior point methods and two types of search directions the hkm and nt directions a discussion of implementation details is provided and computational results on problems from the sdplib and dimacs challenge collections are reported

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

factorization properties of optimum space time processors in nonstationary environments a structural analysis of space time log likelihood processors llp that applies to arbitrary signal transmission models consisting of n sources m sensors a time varying linear channel and nonstationary gaussian source signal and sensor noise processes is presented the approach is based on representing the time varying linear channel as a bounded linear operator l with closed range by exploiting the properties of such operators and the specific structure of the array covariance function it is shown that the classical m dimensional integral equations defining the llp can be transformed into equivalent n dimensional integral equations as a result it is always possible to factor the llp into a cascade of three specialized time varying subprocessors namely a space time whitening filter an m input n output unitary beamformer ub and an n input quadratic postprocessor qpp both the ub and the qpp are given an interpretation and their most important features are indicated to illustrate the theory several examples of its application to signal models involving time varying delays are given

a theoretical study of stein s covariance estimator the traditional covariance estimator the sample covariance matrix which is also the mle is known to be a poor estimator unless the sample size is much larger than the dimension of the covariance matrix stein s estimator has often been regarded as a much better alternative to the mle in small sample sizes and is traditionally used with an isotonizing algorithm the purpose of which is to retain positivity and the original order of the sample eigenvalues despite the superior performance of stein s isotonized estimator in numerical investigations its theoretical properties have not been explored in detail and important questions still remain unanswered one particular question of interest is to identify the regimes under which stein s estimator is guaranteed to perform well a second goal is to determine the extent to which the performance of stein s estimator depends on the isotonizing algorithm the presence of the ad hoc isotonizing algorithm however renders a theoretical analysis rather difficult and consequently risk functions are not easily quantifiable for comparison purposes hence formal decision theoretical results are difficult to obtain and have been elusive ever since the estimator was introduced despite these hurdles in this paper we show that an analysis of stein s covariance estimator within the unbiased estimator of risk ubeor framework can nevertheless lead to important theoretical and methodological insights that are relevant for applications our analysis demonstrates that stein s estimator may give only modest risk reductions when it is not isotonized and when it is isotonized the risk reductions can be significant in particular three broad regimes are identified regarding the behavior of stein s ubeor the theoretical insights are then affirmed at the level of risk functions via numerical simulations

optimal shrinkage of eigenvalues in the spiked covariance model we show that in a common high dimensional covariance model the choice of loss function has a profound effect on optimal estimation in an asymptotic framework based on the spiked covariance model and use of orthogonally invariant estimators we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker that acts elementwise on the sample eigenvalues indeed to each loss function there corresponds a unique admissible eigenvalue shrinker dominating all other shrinkers the shape of the optimal shrinker is determined by the choice of loss function and crucially by inconsistency of both eigenvalues and eigenvectors of the sample covariance matrix details of these phenomena and closed form formulas for the optimal eigenvalue shrinkers are worked out for a menagerie of 26 loss functions for covariance estimation found in the literature including the stein entropy divergence frechet bhattacharya matusita frobenius norm operator norm nuclear norm and condition number losses

a survey of the s lemma in this survey we review the many faces of the s lemma a result about the correctness of the s procedure the basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization convex geometry and linear algebra as well these were all active research areas but as there was little interaction between researchers in these different areas their results remained mainly isolated here we give a unified analysis of the theory by providing three different proofs for the s lemma and revealing hidden connections with various areas of mathematics we prove some new duality results and present applications from control theory error estimation and computational geometry

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

wishart distributions for decomposable graphs when considering a graphical gaussian model n g markov with respect to a decomposable graph g the parameter space of interest for the precision parameter is the cone p g of positive definite matrices with fixed zeros corresponding to the missing edges of g the parameter space for the scale parameter of n g is the cone q g dual to p g of incomplete matrices with submatrices corresponding to the cliques of g being positive definite in this paper we construct on the cones q g and p g two families of wishart distributions namely the type i and type ii wisharts they can be viewed as generalizations of the hyper wishart and the inverse of the hyper inverse wishart as defined by dawid and lauritzen ann statist 21 1993 1272 1317 we show that the type i and ii wisharts have properties similar to those of the hyper and hyper inverse wishart indeed the inverse of the type ii wishart forms a conjugate family of priors for the covariance parameter of the graphical gaussian model and is strong directed hyper markov for every direction given to the graph by a perfect order of its cliques while the type i wishart is weak hyper markov moreover the inverse type ii wishart as a conjugate family presents the advantage of having a multidimensional shape parameter thus offering flexibility for the choice of a prior both type i and ii wishart distributions depend on multivariate shape parameters a shape parameter is acceptable if and only if it satisfies a certain eigenvalue property we show that the sets of acceptable shape parameters for a noncomplete g have dimension equal to at least one plus the number of cliques in g these families as conjugate families are richer than the traditional diaconis ylvisaker conjugate families which all have a shape parameter set of dimension one a decomposable graph which does not contain a three link chain as an induced subgraph is said to be homogeneous in this case our wisharts are particular cases of the wisharts on homogeneous cones as defined by andersson and wojnar j theoret probab 17 2004 781 818 and the dimension of the shape parameter set is even larger than in the nonhomogeneous case it is indeed equal to the number of cliques plus the number of distinct minimal separators using the model where g is a three link chain we show by computing a 7 tuple integral that in general we cannot expect the shape parameter sets to have dimension larger than the number of cliques plus one

worst case conditional value at risk with application to robust portfolio management this paper considers the worst case conditional value at risk cvar in the situation where only partial information on the underlying probability distribution is available the minimization of the worst case cvar under mixture distribution uncertainty box uncertainty and ellipsoidal uncertainty are investigated the application of the worst case cvar to robust portfolio optimization is proposed and the corresponding problems are cast as linear programs and second order cone programs that can be solved efficiently market data simulation and monte carlo simulation examples are presented to illustrate the proposed approach

a robust optimization perspective on stochastic programming in this paper we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for random variables termed the forward and backward deviations these deviation measures capture distributional asymmetry and lead to better approximations of chance constraints using a linear decision rule we also propose a tractable approximation approach for solving a class of multistage chance constrained stochastic linear optimization problems an attractive feature of the framework is that we convert the original model into a second order cone program which is computationally tractable both in theory and in practice we demonstrate the framework through an application of a project management problem with uncertain activity completion time

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

on the condition number anomaly of gaussian correlation matrices abstract spatial correlation matrices appear in a large variety of applications for example they are an essential component of spatial gaussian processes also known as spatial linear models or kriging estimators which are powerful and well established tools for a multitude of engineering applications such as the design and analysis of computer experiments geostatistical problems and meteorological tasks in radial basis function interpolation gaussian correlation matrices arise frequently as interpolation matrices from the gaussian radial kernel function in the field of data assimilation in numerical weather prediction such matrices arise as background error covariances over the past thirty years it was observed by several authors from several fields that the gaussian correlation model is exceptionally prone to suffer from ill conditioning but a quantitative theoretical explanation for this anomaly was lacking in this paper a proof for the special position of the gaussian correlation matrix is given the theoretical findings are illustrated by numerical experiment

on the computation of weighted analytic centers and dual ellipsoids with the projective algorithm the primal projective algorithm for linear programs with unknown optimal objective function value is extended to the case where one uses a weighted karmarkar potential function this potential is defined with respect to a strict lower bound to the optimum the minimization of this potential when the lower bound is kept fixed yields a primal and a dual feasible solution the dual solution is the weighted analytic center of a certain dual polytope finally one exhibits a pair of homothetic dual ellipsoids that extends results obtained by sonnevend todd ye freund and anstreicher

robust mean covariance solutions for stochastic optimization we provide a method for deriving robust solutions to certain stochastic optimization problems based on mean covariance information about the distributions underlying the uncertain vector of returns we prove that for a general class of objective functions the robust solutions amount to solving a certain deterministic parametric quadratic program we first prove a general projection property for multivariate distributions with given means and covariances which reduces our problem to optimizing a univariate mean variance robust objective this allows us to use known univariate results in the multidimensional setting and to add new results in this direction in particular we characterize a general class of objective functions the so called one or two point support functions for which the robust objective is reduced to a deterministic optimization problem in one variable finally we adapt a result from geoffrion 1967a to reduce the main problem to a parametric quadratic program in particular our results are true for increasing concave utilities with convex or concave convex derivatives closed form solutions are obtained for special discontinuous criteria motivated by bonus and commission based incentive schemes for portfolio management we also investigate a multiproduct pricing application which motivates extensions of our results for the case of nonnegative and decision dependent returns

linear time algorithms for two and three variable linear programs o n time algorithms for linear programming problems with two or three variables and n constraints are described the approach uses convexity dominance of linear functions and linear time median finding algorithms the algorithms improve the previously known best bounds of o n log n time for both of these problems

a cutting surface algorithm for semi infinite convex programming with an application to moment robust optimization we present and analyze a central cutting surface algorithm for general semi infinite convex optimization problems and use it to develop a novel algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments moments of arbitrary order as well as nonpolynomial moments can be included in the formulation we show that this gives rise to a hierarchy of optimization problems with decreasing levels of risk aversion with classic robust optimization at one end of the spectrum and stochastic programming at the other although our primary motivation is to solve distributionally robust optimization problems with moment uncertainty the cutting surface method for general semi infinite convex programs is also of independent interest the proposed method is applicable to problems with nondifferentiable semi infinite constraints indexed by an infinite dimensional index set examples comparing the cutting surface algorit

localization of narrowband radio emitters based on doppler frequency shifts several techniques for emitter localization based on the doppler effect have been described in the literature one example is the differential doppler dd method in which the signal of a stationary emitter is intercepted by at least two moving receivers the frequency difference between the receivers is measured at several locations along their trajectories and the emitter s position is then estimated based on these measurements this two step approach is suboptimal since each frequency difference measurement is performed independently although all measurements correspond to a common emitter position instead a single step approach based on the maximum likelihood criterion is proposed here for both known and unknown waveforms the position is determined directly from all the observations by a search in the position space the method can only be used for narrowband signals that is under the assumption that the signal bandwidth must be small compared to the inverse of the propagation time between the receivers simulations show that the proposed method outperforms the dd method for weak signals while both methods converge to the cramer rao bound for strong known signals finally it is shown that in some cases of interest the proposed method inherently selects reliable observations while ignoring unreliable data

data driven distributionally robust optimization using the wasserstein metric performance guarantees and tractable reformulations we consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset using the wasserstein metric we construct a ball in the space of multivariate and non discrete probability distributions centered at the uniform distribution on the training samples and we seek decisions that perform best in view of the worst case distribution within this wasserstein ball the state of the art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques which quickly become computationally excruciating in this paper we demonstrate that under mild assumptions the distributionally robust optimization problems over wasserstein balls can in fact be reformulated as finite convex programs in many interesting cases even as tractable linear programs leveraging recent measure concentration results we also show that their solutions enjoy powerful finite sample performance guarantees our theoretical results are exemplified in mean risk portfolio optimization as well as uncertainty quantification

multidimensional scaling analysis for passive moving target localization with tdoa and fdoa measurements a new framework for positioning a moving target is introduced by utilizing time differences of arrival tdoa and frequency differences of arrival fdoa measurements collected using an array of passive sensors it exploits the multidimensional scaling mds analysis which has been developed for data analysis in the field such as physics geography and biology particularly we present an accurate and closed form solution for the position and velocity of a moving target unlike most passive target localization methods focusing on minimizing a loss function with respect to the measurement vector the proposed method is based on the optimization of a cost function related to the scalar product matrix in the classical mds framework it is robust to the large measurement noise the bias and variance of the proposed estimator is also derived simulation results show that the proposed estimator achieves better performance than the spherical interpolation si method and the two step weighted least squares wls approach and it attains the cramer rao lower bound at a sufficiently high noise level before the threshold effect occurs moreover for the proposed estimator the threshold effect which is a result of the nonlinear nature of the localization problem occurs apparently later as the measurement noise increases for a near field target

model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data we consider the problem of estimating the parameters of a gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse our approach is to solve a maximum likelihood problem with an added l1 norm penalty term the problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes we present two new algorithms for solving problems with at least a thousand nodes in the gaussian case our first algorithm uses block coordinate descent and can be interpreted as recursive l1 norm penalized regression our second algorithm based on nesterov s first order method yields a complexity estimate with a better dependence on problem size than existing interior point methods using a log determinant relaxation of the log partition function wainwright and jordan 2006 we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case we test our algorithms on synthetic data as well as on gene expression and senate voting records data