a bayesian approach to adaptive detection in nonhomogeneous environments we consider the adaptive detection of a signal of interest embedded in colored noise when the environment is nonhomogeneous i e when the training samples used for adaptation do not share the same covariance matrix as the vector under test a bayesian framework is proposed where the covariance matrices of the primary and the secondary data are assumed to be random with some appropriate joint distribution the prior distributions of these matrices require a rough knowledge about the environment this provides a flexible yet simple knowledge aided model where the degree of nonhomogeneity can be tuned through some scalar variables within this framework an approximate generalized likelihood ratio test is formulated accordingly two bayesian versions of the adaptive matched filter are presented where the conventional maximum likelihood estimate of the primary data covariance matrix is replaced either by its minimum mean square error estimate or by its maximum a posteriori estimate two detectors require generating samples distributed according to the joint posterior distribution of primary and secondary data covariance matrices this is achieved through the use of a gibbs sampling strategy numerical simulations illustrate the performances of these detectors and compare them with those of the conventional adaptive matched filter

time varying autoregressive moving average models for covariance estimation we consider large scale covariance estimation using a small number of samples in applications where there is a natural ordering between the random variables the two classical approaches to this problem rely on banded covariance and banded inverse covariance structures corresponding to time varying moving average ma and autoregressive ar models respectively motivated by this analogy to spectral estimation and the well known modeling power of autoregressive moving average arma processes we propose a novel time varying arma covariance structure similarly to known results in the context of ar and ma we address the completion of an arma covariance matrix from its main band and its estimation based on random samples finally we examine the advantages of our proposed methods using numerical experiments

covariance matrix estimation with heterogeneous samples we consider the problem of estimating the covariance matrix mp of an observation vector using heterogeneous training samples i e samples whose covariance matrices are not exactly mp more precisely we assume that the training samples can be clustered into k groups each one containing lk snapshots sharing the same covariance matrix mk furthermore a bayesian approach is proposed in which the matrices mk are assumed to be random with some prior distribution we consider two different assumptions for mp in a fully bayesian framework mp is assumed to be random with a given prior distribution under this assumption we derive the minimum mean square error mmse estimator of mp which is implemented using a gibbs sampling strategy moreover a simpler scheme based on a weighted sample covariance matrix scm is also considered the weights minimizing the mean square error mse of the estimated covariance matrix are derived furthermore we consider estimators based on colored or diagonal loading of the weighted scm and we determine theoretically the optimal level of loading finally in order to relax the a priori assumptions about the covariance matrix mp the second part of the paper assumes that this matrix is deterministic and derives its maximum likelihood estimator numerical simulations are presented to illustrate the performance of the different estimation schemes

the tortoise and the hare restart gmres when solving large nonsymmetric systems of linear equations with the restarted gmres algorithm one is inclined to select a relatively large restart parameter in the hope of mimicking the full gmres process surprisingly cases exist where small values of the restart parameter yield convergence in fewer iterations than larger values here two simple examples are presented where gmres 1 converges exactly in three iterations while gmres 2 stagnates one of these examples reveals that gmres 1 convergence can be extremely sensitive to small changes in the initial residual

the emergence of the electrostatic field as a feynman sum in random tilings with holes we consider random lozenge tilings on the triangular lattice with holes q 1 q n in some fixed position for each unit triangle not in a hole consider the average orientation of the lozenge covering it we show that the scaling limit of this discrete field is the electrostatic field obtained when regarding each hole q i as an electrical charge of magnitude equal to the difference between the number of unit triangles of the two different orientations inside q i this is then restated in terms of random surfaces yielding the result that the average over surfaces with prescribed height at the union of the boundaries of the holes is in the scaling limit a sum of helicoids

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

radar detection of distributed targets in homogeneous interference whose inverse covariance structure is defined via unitary invariant functions in this paper we deal with the problem of detecting an extended target embedded in homogeneous gaussian interference with unknown but structured covariance matrix we model the possible target echo from each range bin under test as a deterministic signal with an unknown scaling factor accounting for the target response at the design stage we exploit some a priori knowledge about the operating environment enforcing the inverse interference plus noise covariance matrix to belong to a set described via unitary invariant continuous functions hence we derive the constrained maximum likelihood ml estimates of the unknown parameters under both the h0 and h1 hypotheses and design the generalized likelihood ratio test glrt for the considered decision problem at the analysis stage we assess the performance of the devised glrt for some covariance matrix uncertainty sets of practical relevance both for spatial and doppler processing the results highlight that correct use of the a priori knowledge can lead to a detection performance quite close to the optimum receiver which supposes the perfect knowledge of the interference plus noise covariance matrix

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

performance analysis of tyler s covariance estimator this paper analyzes the performance of tyler s m estimator of the scatter matrix in elliptical populations we focus on the non asymptotic setting and derive estimation error bounds depending on the number of samples n and the dimension p we show that under mild conditions the squared frobenius norm of the error of the inverse estimator decays like p 2 n with high probability

joint estimation of inverse covariance matrices lying in an unknown subspace we consider the problem of joint estimation of inverse covariance matrices lying in an unknown subspace of the linear space of symmetric matrices we perform the estimation using groups of measurements with different covariances assuming the inverse covariances span a low dimensional subspace our aim is to determine this subspace and to exploit this knowledge in order to improve the estimation we develop a novel optimization algorithm discovering and exploiting the underlying low dimensional subspace we provide a computationally efficient algorithm and derive a tight upper performance bound numerical simulations on synthetic and real world data are presented to illustrate the performance benefits of the algorithm

macroscopically separated gaps in dimer coverings of aztec rectangles in this paper we determine the interaction of diagonal defect clusters in regions of an aztec rectangle that scale to arbitrary points on its symmetry axis in earlier work we treated the case when this point was the center of the scaled aztec rectangle we use the resulting formulas to determine the asymptotics of the correlation of defects that are macroscopically separated from one another and feel the influence of the boundary in several of the treated situations this seems not to be accomplishable by previous methods our applications include the case of two long neutral strings which turn out to interact by an analog of the casimir force two families of neutral doublets that turn out to interact completely independently of one another a neutral doublet and a very long neutral string a general collection of macroscopically separated monomer and separation defects and the case of long strings consisting of consecutive monomers

ritz values of normal matrices and ceva s theorem abstract this paper investigates the behavior of ritz values of normal matrices we apply ceva s theorem a classical geometric result to understand the geometric relationship between pairs of ritz values for 3 3 normal non hermitian matrices and then analyze the implications for larger matrices we find that in the case of normal non hermitian matrices the geometric constraints on the placement of ritz values provide less freedom than the cauchy interlacing theorem in the hermitian case using our results we analyze the restarted arnoldi method with exact shifts applied to a 3 3 normal non hermitian matrix

better subset regression using the nonnegative garrote a new method called the nonnegative nn garrote is proposed for doing subset regression it both shrinks and zeroes coefficients in tests on real and simulated data it produces lower prediction error than ordinary subset selection it is also compared to ridge regression if the regression equations generated by a procedure do not change drastically with small changes in the data the procedure is called stable subset selection is unstable ridge is very stable and the nn garrote is intermediate simulation results illustrate the effects of instability on prediction error

embedding nonnegative definite toeplitz matrices in nonnegative definite circulant matrices with application to covariance estimation the class of nonnegative definite toeplitz matrices that can be embedded in nonnegative definite circulant matrices of a larger size is characterized an equivalent characterization in terms of the spectrum of the underlying process is also presented together with the corresponding extremal processes it is shown that a given finite duration sequence rho can be extended to be the covariance of a periodic stationary processes whenever the toeplitz matrix r generated by this sequence is strictly positive definite the sequence rho 1 cos alpha cos 2 alpha with alpha pi irrational which has a unique nonperiodic extension as a covariance sequence demonstrates that the strictness is needed a simple constructive proof supplies a bound on the abovementioned period in terms of the minimal eigenvalue of r it also yields under the same conditions an extension of rho to covariances that eventually decay to zero for the maximum likelihood estimate of the covariance of a stationary gaussian process the extension length required for using the estimate maximize iterative algorithm is determined

joint estimation of multiple precision matrices with common structures estimation of inverse covariance matrices known as precision matrices is important in various areas of statistical analysis in this article we consider estimation of multiple precision matrices sharing some common structures in this setting estimating each precision matrix separately can be suboptimal as it ignores potential common structures this article proposes a new approach to parameterize each precision matrix as a sum of common and unique components and estimate multiple precision matrices in a constrained l1 minimization framework we establish both estimation and selection consistency of the proposed estimator in the high dimensional setting the proposed estimator achieves a faster convergence rate for the common structure in certain cases our numerical examples demonstrate that our new estimator can perform better than several existing methods in terms of the entropy loss and frobenius loss an application to a glioblastoma cancer data set reveals some interesting gene networks across multiple cancer subtypes

imbedding conditions for normal matrices abstract when can an n k n k normal matrix b be imbedded in an n n normal matrix a this question was studied for the first time 50 years ago by ky fan and gordon pall who gave the complete answer in the case k 1 since then a few authors obtained additional results in this note we show how an approach inspired by the hermitian case can throw some light on the problem

interaction between hermitian and normal imbeddings abstract for several applications it is highly desirable to understand how the eigenvalues of an imbeddable matrix v av where v c n k is an isometry are distributed throughout the numerical range of a c n n there has been extensive study for a hermitian while a geometric description for the eigenvalues of imbeddings in non hermitian matrices remains a challenging problem toward this direction a subspace is introduced wherein all n n complex diagonal matrices for which a given isometry v c n k generates diagonal imbeddings are defined in particular conditions upon which a real diagonal matrix may be imbeddable in some normal are obtained including an application for higher rank numerical ranges finally a procedure determining whether two given sets of complex numbers may be realized as spectra of a pair of imbeddable normal matrices is established

hypercomplex correlation techniques for vector images correlation techniques have been applied to almost every area of signal processing over the past century yet their use has in general been limited to scalar signals while there have been implementations in multichannel applications these can be characterized as a combination of single channel processes true vector correlation techniques with global and interchannel measures have only recently been demonstrated and are still in their infancy by comparison this paper describes our work on vector correlation based on the use of hypercomplex fourier transforms and presents for the first time a unified theory behind the information contained in the peak of a vector correlation response by using example applications for color images we also demonstrate some of the practical implications together with our latest results

dimers and amoebae we study random surfaces which arise as height functions of random perfect matchings a k a dimer configurations on a weighted bipartite doubly periodic graph g embedded in the plane we derive explicit formulas for the surface tension and local gibbs measure probabilities of these models the answers involve a certain plane algebraic curve which is the spectral curve of the kasteleyn operator of the graph for example the surface tension is the legendre dual of the ronkin function of the spectral curve the amoeba of the spectral curve represents the phase diagram of the dimer model further we prove that the spectral curve of a dimer model is always a real curve of special type namely it is a harnack curve this implies many qualitative and quantitative statement about the behavior of the dimer model such as existence of smooth phases decay rate of correlations growth rate of height function fluctuations etc

discrete fractal dimensions of the ranges of random walks in mathbb z d associate with random conductances let x x t t ge 0 be a continuous time random walk in an environment of i i d random conductances mu e in 1 infty e in e d where e d is the set of nonoriented nearest neighbor bonds on the euclidean lattice mathbb z d and d 3 let rm r x in mathbb z d x t x rm for some t ge 0 be the range of x it is proved that for almost every realization of the environment dimh r dimp r 2 almost surely where dimh and dimp denote respectively the discrete hausdorff and packing dimension furthermore given any set a subseteq mathbb z d a criterion for a to be hit by x t for arbitrarily large t 0 is given in terms of dimh a similar results for bouchoud s trap model in mathbb z d d 3 are also proven

covariance regularized regression and classification for high dimensional problems in recent years many methods have been developed for regression in high dimensional settings we propose covariance regularized regression a family of methods that use a shrunken estimate of the inverse covariance matrix of the features in order to achieve superior prediction an estimate of the inverse covariance matrix is obtained by maximizing its log likelihood under a multivariate normal model subject to a constraint on its elements this estimate is then used to estimate coefficients for the regression of the response onto the features we show that ridge regression the lasso and the elastic net are special cases of covariance regularized regression and we demonstrate that certain previously unexplored forms of covariance regularized regression can outperform existing methods in a range of situations the covariance regularized regression framework is extended to generalized linear models and linear discriminant analysis and is used to analyze gene expression data sets with multiple class and survival outcomes

on recursive estimation for time varying autoregressive processes this paper focuses on recursive estimation of time varying autoregressive processes in a nonparametric setting the stability of the model is revisited and uniform results are provided when the time varying autoregressive parameters belong to appropriate smoothness classes an adequate normalization for the correction term used in the recursive estimation procedure allows for very mild assumptions on the innovations distributions the rate of convergence of the pointwise estimates is shown to be minimax in lipschitz classes for 0 1 for 1 2 this property no longer holds this can be seen by using an asymptotic expansion of the estimation error a bias reduction method is then proposed for recovering the minimax rate

inverse spectral problem for normal matrices and the gauss lucas theorem we establish an analog of the cauchy poincare interlacing theorem for normal matrices in terms of majorization and we provide a solution to the corresponding inverse spectral problem using this solution we generalize and extend the gauss lucas theorem and prove the old conjecture of de bruijn springer on the location of the roots of a complex polynomial and its derivative and an analog of rolle s theorem conjectured by schoenberg

optimal rates of convergence for estimating toeplitz covariance matrices toeplitz covariance matrices are used in the analysis of stationary stochastic processes and a wide range of applications including radar imaging target detection speech recognition and communications systems in this paper we consider optimal estimation of large toeplitz covariance matrices and establish the minimax rate of convergence for two commonly used parameter spaces under the spectral norm the properties of the tapering and banding estimators are studied in detail and are used to obtain the minimax upper bound the results also reveal a fundamental difference between the tapering and banding estimators over certain parameter spaces the minimax lower bound is derived through a novel construction of a more informative experiment for which the minimax lower bound is obtained through an equivalent gaussian scale model and through a careful selection of a finite collection of least favorable parameters in addition optimal rate of convergence for estimating the inverse of a toeplitz covariance matrix is also established

time varying autoregressive tvar models for multiple radar observations we consider the adaptive radar problem where the properties of the nonstationary clutter signals can be estimated using multiple observations of radar returns from a number of sufficiently homogeneous range azimuth resolution cells we derive a method for approximating an arbitrary hermitian covariance matrix by a time varying autoregressive model of order m tvar m that is based on the dym gohberg band matrix extension technique which gives the unique tvar m model for any nondegenerate covariance matrix we demonstrate that the dym gohberg transformation of the sample covariance matrix gives the maximum likelihood ml estimate of the tvar m covariance matrix we introduce an example of tvar m clutter modeling for high frequency over the horizon radar that demonstrates its practical importance

on the linear convergence of the alternating direction method of multipliers we analyze the convergence rate of the alternating direction method of multipliers admm for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints previous analysis of the admm typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks moreover there has been no rate of convergence analysis for the admm without strong convexity in the objective function in this paper we establish the global r linear convergence of the admm for minimizing the sum of any number of convex separable functions assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping and a nonsmooth ell 1 l1 regularizer this result implies the linear convergence of the admm for contemporary applications such as lasso without assuming strong convexity of the objective function

a comparison between hmlp and hrbf for attitude control in this paper the problem of controlling the attitude of a rigid body such as a spacecraft in three dimensional space is approached by introducing two new control strategies developed in hypercomplex algebra the proposed approaches are based on two parallel controllers both derived in quaternion algebra the first is a feedback controller of the proportional derivative pd type while the second is a feedforward controller which is implemented either by means of a hypercomplex multilayer perceptron hmlp neural network or by means of a hypercomplex radial basis function hrbf neural network several simulations show the performance of the two approaches the results are also compared with a classical pd controller and with an adaptive controller showing the improvements obtained by using neural networks especially when an external disturbance acts on the rigid body in particular the hmlp network gave better results when considering trajectories not presented during the learning phase