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

the bernstein orlicz norm and deviation inequalities we introduce two new concepts designed for the study of empirical processes first we introduce a new orlicz norm which we call the bernstein orlicz norm this new norm interpolates sub gaussian and sub exponential tail behavior in particular we show how this norm can be used to simplify the derivation of deviation inequalities for suprema of collections of random variables secondly we introduce chaining and generic chaining along a tree these simplify the well known concepts of chaining and generic chaining the supremum of the empirical process is then studied as a special case we show that chaining along a tree can be done using entropy with bracketing finally we establish a deviation inequality for the empirical process for the unbounded case

the sinc method in multiple space dimensions model problems the sinc galerkin method is applied to the canonical forms of second order partial differential equations in multiple space dimensions for time dependent problems the scheme is fully galerkin i e the domain of the basis elements includes time hence the approximate solution for each equation is specified by solving the associated linear system notation is developed which facilitates description of both the linear systems and the algorithms used to solve them alternative algorithms are offered dependent on a scalar versus vector computing environment numerical results for the test problems presented sustain the exponential convergence rate of the method even in the presence of singularities

binary n step markov chains and long range correlated systems a theory of systems with long range correlations based on the consideration of binary n step markov chains is developed in our model the conditional probability that the i th symbol in the chain equals zero or unity is a linear function of the number of unities among the preceding n symbols the correlation and distribution functions as well as the variance of number of symbols in the words of arbitrary length l are obtained analytically and numerically if the persistent correlations are not extremely strong the variance is shown to be nonlinearly dependent on l a self similarity of the studied stochastic process is revealed the applicability of the developed theory to the coarse grained written and dna texts is discussed

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 joint graphical lasso for inverse covariance estimation across multiple classes we consider the problem of estimating multiple related gaussian graphical models from a high dimensional data set with observations belonging to distinct classes we propose the joint graphical lasso which borrows strength across the classes in order to estimate multiple graphical models that share certain characteristics such as the locations or weights of nonzero edges our approach is based upon maximizing a penalized log likelihood we employ generalized fused lasso or group lasso penalties and implement a fast admm algorithm to solve the corresponding convex optimization problems the performance of the proposed method is illustrated through simulated and real data examples

on the early history of the singular value decomposition this paper surveys the contributions of five mathematicians eugenio beltrami 1835 1899 camille jordan 1838 1921 james joseph sylvester 1814 1897 erhard schmidt 1876 1959 and hermann weyl 1885 1955 who were responsible for establishing the existence of the singular value decomposition and developing its theory

optshrink an algorithm for improved low rank signal matrix denoising by optimal data driven singular value shrinkage the truncated singular value decomposition of the measurement matrix is the optimal solution to the representation problem of how to best approximate a noisy measurement matrix using a low rank matrix here we consider the unobservable denoising problem of how to best approximate a low rank signal matrix buried in noise by optimal re weighting of the singular vectors of the measurement matrix we exploit recent results from random matrix theory to exactly characterize the large matrix limit of the optimal weighting coefficients and show that they can be computed directly from data for a large class of noise models that includes the independent identically distributed gaussian noise case our analysis brings into sharp focus the shrinkage and thresholding form of the optimal weights the nonconvex nature of the associated shrinkage function on the singular values and explains why matrix regularization via singular value thresholding with convex penalty functions such as the nuclear norm will always be suboptimal we validate our theoretical predictions with numerical simulations develop an implementable algorithm optshrink that realizes the predicted performance gains and show how our methods can be used to improve estimation in the setting where the measured matrix has missing entries

matrix decomposition algorithms in orthogonal spline collocation for separable elliptic boundary value problems fast direct methods are presented for the solution of linear systems arising in high order tensor product orthogonal spline collocation applied to separable second order linear elliptic partial

quasi trefftz spectral method for separable linear elliptic equations abstract a new numerical method qtsm which was earlier suggested for solving boundary value problems with elliptic operators with constant coefficients is applied to separable second order elliptic equations with varying coefficients the problems in irregular domains are considered the method combines the properties of the boundary methods with the spectral representation of the solution in the form of expansion over the eigenfunctions of some sturm liouville problem the method is tested on several one and two dimensional problems with exact analytic solution the possibilities of further developments of the method are discussed

lower bounds on the mean squared error of low rank matrix reconstruction we investigate the behavior of the mean square error mse of low rank matrix reconstruction and its special case matrix completion we first derive the constrained cramer rao bound crb on the mse matrix of any locally unbiased estimator and then analyze the behavior of the constrained crb when a subset of entries of the underlying matrix is randomly observed we design an alternating minimization procedure to compute the maximum likelihood estimator mle for the low rank matrix and demonstrate through numerical simulations that the performance of the mle approaches the constrained crb when the signal to noise ratio is high applying a chapman robbins type barankin bound allows us to derive lower bounds on the worst case scalar mse we demonstrate that the worst case scalar mse is infinite even if the model is identifiable however the infinite scalar mse is achieved only on a set of low rank matrices with measure zero we discuss the implications of these bounds and compare them with the empirical performance of the matrix lasso estimator and the existing bounds in the literature

approximation of multivariate functions and evaluation of particular solutions using chebyshev polynomial and trigonometric basis functions summary a two stage numerical procedure using chebyshev polynomials and trigonometric functions is proposed to approximate the source term of a given partial differential equation the purpose of such numerical schemes is crucial for the evaluation of particular solutions of a large class of partial differential equations our proposed scheme provides a highly efficient and accurate approximation of multivariate functions and particular solution of certain partial differential equations simultaneously numerical results on the approximation of eight two dimensional test functions and their derivatives are given to demonstrate that the scheme for the approximation of functions can be easily extended to evaluate the particular solution of certain partial differential equations we solve a modified helmholtz equation near machine precision can be achieved for all these test problems copyright 2006 john wiley sons ltd

multi directed eulerian growing networks abstract we introduce and analyze a model of a multi directed eulerian network that is a directed and weighted network where a path exists that passes through all the edges of the network once and only once networks of this type can be used to describe information networks such as human language or dna chains we are able to calculate the strength and degree distribution in this network and find that they both exhibit a power law with an exponent between 2 and 3 we then analyze the behavior of the accelerated version of the model and find that the strength distribution has a double slope power law behavior finally we introduce a non eulerian version of the model and find that the statistical topological properties remain unchanged our analytical results are compared with numerical simulations

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

a distribution free m estimator of multivariate scatter the existence and uniqueness of a limiting form of a huber type m estimator of multivariate scatter is established under certain conditions on the observed sample these conditions hold with probability one when sampling randomly from a continuous multivariate distribution the existence of the estimator is proven by showing that it is the limiting point of a specific algorithm hence the proof is constructive for continuous populations the estimator of multivariate scatter is shown to be strongly consistent and asymptotically normal an important property of the estimator is that its asymptotic distribution is distribution free with respect to the class of continuous elliptically distributed populations this distribution free property also holds for the finite sample size distribution when the location parameter is known in addition the estimator is the most robust estimator of the scatter matrix of an elliptical distribution in the sense of minimizing the maximum asymptotic variance

adaptive estimation of a quadratic functional by model selection we consider the problem of estimating s 2 when s belongs to some separable hilbert space and one observes the gaussian process y t langles t rangle sigmal t for all t epsilon mathbb h where l is some gaussian isonormal process this framework allows us in particular to consider the classical gaussian sequence model for which mathbb h l 2 mathbb n and l t sum lambda geq1 t lambda varepsilon lambda where varepsilon lambda lambda geq1 is a sequence of i i d standard normal variables our approach consists in considering some at most countable families of finite dimensional linear subspaces of mathbb h the models and then using model selection via some conveniently penalized least squares criterion to build new estimators of s 2 we prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s depending on the family of models from which they are built in particular in the context of the gaussian sequence model a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles ellipsoids l p bodies or besov bodies we take special care to describe the conditions under which the penalized estimator is efficient when the level of noise sigma tends to zero our construction is an alternative to the one by efroimovich and low for hyperrectangles and provides new results otherwise

sinc function quadrature rules for the fourier integral in this paper a numerical algorithm is proposed for the computation of the fourier transform the quadrature rule developed is based on the whittaker cardinal function expansion of the integrand and a certain conformal map the error of the method is analyzed and numerical results are reported which confirm the accuracy of the quadrature rule

an interpretation of the moore penrose generalized inverse of a singular fisher information matrix it is proved that in a non bayesian parametric estimation problem if the fisher information matrix fim is singular unbiased estimators for the unknown parameter will not exist cramer rao bound crb a popular tool to lower bound the variances of unbiased estimators seems inapplicable in such situations in this correspondence we show that the moore penrose generalized inverse of a singular fim can be interpreted as the crb corresponding to the minimum variance among all choices of minimum constraint functions this result ensures the logical validity of applying the moore penrose generalized inverse of an fim as the covariance lower bound when the fim is singular furthermore the result can be applied as a performance bound on the joint design of constraint functions and unbiased estimators

block matrices with l block banded inverse inversion algorithms block banded matrices generalize banded matrices we study the properties of positive definite full matrices p whose inverses a are l block banded we show that for such matrices the blocks in the l block band of p completely determine p namely all blocks of p outside its l block band are computed from the blocks in the l block band of p we derive fast inversion algorithms for p and its inverse a that when compared to direct inversion are faster by two orders of magnitude of the linear dimension of the constituent blocks we apply these inversion algorithms to successfully develop fast approximations to kalman bucy filters in applications with high dimensional states where the direct inversion of the covariance matrix is computationally unfeasible

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

sinc type approximations in h 1 norm with applications to boundary value problems abstract sinc functions are used for approximation on a general arc in the complex plane it is proved that for certain functions analytic in a domain containing the arc the error of an n term approximation converges to zero in the sobolev space h1 at the rate o e n 1 2 as n special attention is given to the important case of a b where a and b are finite real numbers an application is also considered of the approximate result to the numerical solution of two point boundary value problems it is shown that the classical sinc galerkin method with n basis functions has o e n 1 2 convergence rate in the h1 norm

high order fast direct methods for separable elliptic equations the rayleigh ritz galerkin method with tensor product b splines yields high order discretiz ations for elliptic partial differential equations for smooth problems the resulting linear system of equations is both smaller and denser than the corresponding systems for lower order discretizations however several fast direct methods are known for solving these low order systems when the partial differential equation is separable in this paper we show how to extend the matrix decomposition technique to yield a fast direct method for high order finite element discretizations

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

matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions abstract bialecki b and g fairweather matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions journal of computational and applied mathematics 46 1993 369 386 in recent years several matrix decomposition algorithms have been developed for the efficient solution of the linear algebraic systems arising when finite difference finite element galerkin and spectral methods are applied to separable elliptic boundary value problems in a rectangle the success of these methods depends on knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two point boundary value problems in one space dimension the primary purpose of this paper is to provide an overview of matrix decomposition algorithms and show how they can be expressed in terms of a unifying framework particular emphasis is placed on algorithms formulated recently by the authors for solving the linear systems arising in orthogonal spline collocation that is spline collocation at gauss points all of the methods discussed in this paper are modular and possess a great deal of natural parallelism

sinc galerkin method based on the de transformation for the boundary value problem of fourth order ode in this paper a sinc galerkin method incorporated with the double exponential transformation abbreviated as the de transformation for the two point boundary value problem of fourth order ordinary differential equation is considered in this method the error bound o exp c n logn c 0 is attained as in the sinc collocation method based on the de transformation where n is a parameter representing the number of terms in the sinc approximation high efficiency of the sinc galerkin method with the de transformation is confirmed by some numerical examples and the numerical results were compared with ones obtained by sinc collocation method based on the de transformation

sinc collocation method with orthogonalization for singular poisson like problems this paper uses the sinc collocation method to solve singular poisson like problems a firstor higher order partial derivative of the exact solution is unbounded on the boundary a linear system is obtained which is the same as that obtained by using the sinc galerkin method with a smart choice of the stepsize and the number of the gridpoints the orthogonalization technique is successfully applied to solve the linear system obtained and a numerical approximation is obtained with an exponential accuracy o exp cn7 where n is a truncation parameter and c is a constant independent of n

joint covariance estimation with mutual linear structure we consider the problem of joint estimation of structured covariance matrices assuming the structure is unknown estimation is achieved using heterogeneous training sets namely given groups of measurements coming from centered populations with different covariances our aim is to determine the mutual structure of these covariance matrices and estimate them supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices we develop a new efficient algorithm discovering the structure and using it to improve the estimation our technique is based on the application of principal component analysis in the matrix space we also derive an upper performance bound of the proposed algorithm in the gaussian scenario and compare it with the cramer rao lower bound numerical simulations are presented to illustrate the performance benefits of the proposed method

directed accelerated growth application in citation network abstract in many real world networks the number of links increases non linearly with the number of nodes models of such accelerated growth have been considered earlier with deterministic and stochastic number of links here we consider stochastic accelerated growth in a network where links are directed with the number of outgoing links following a power law distribution the results are similar to the undirected case as the accelerated growth is enhanced the degree of distribution becomes independent of the initial attractiveness a parameter which plays a key role in directed networks as an example of a directed model with accelerated growth the citation network is considered in which the distribution of the number of outgoing link has an exponential tail the role of accelerated growth is examined here with two different growth laws

symmetrization of the sinc galerkin method for boundary value problems on procede a la symetrisation de la methode de galerkin en modifiant la fonction poids pour une classe etendue de problemes aux valeurs limites

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

joint estimation of multiple graphical models gaussian graphical models explore dependence relationships between random variables through the estimation of the corresponding inverse covariance matrices in this paper we develop an estimator for such models appropriate for data from several graphical models that share the same variables and some of the dependence structure in this setting estimating a single graphical model would mask the underlying heterogeneity while estimating separate models for each category does not take advantage of the common structure we propose a method that jointly estimates the graphical models corresponding to the different categories present in the data aiming to preserve the common structure while allowing for differences between the categories this is achieved through a hierarchical penalty that targets the removal of common zeros in the inverse covariance matrices across categories we establish the asymptotic consistency and sparsity of the proposed estimator in the high dimensional case and illustrate its performance on a number of simulated networks an application to learning semantic connections between terms from webpages collected from computer science departments is included

matrix decomposition algorithms for arbitrary order c 0 tensor product finite element systems matrix decomposition algorithms mdas are fast direct methods for the solution of systems of linear algebraic equations which arise in the approximation of poisson s equation on the unit square using various techniques such as finite difference spline collocation and spectral methods the attraction of mdas is that they employ fast fourier transforms and require o n 2 log n operations on an n n uniform partition of the unit square in this paper mdas are formulated for the solution of the finite element galerkin equations arising when spaces of c 0 piecewise polynomials of degree k 3 are employed results of numerical experiments exhibit the expected optimal global convergence rates and superconvergence phenomena

the topology of an accelerated growth network we present and investigate a general nonlinear growth network model which incorporates accelerated growth of nodes and edges where the growth rates of edges and nodes are all time dependent power law functions the acceleration of edges determines the proportion of the internal edges to the external edges which play a key role influencing the structure of the network on the other hand the effects of the acceleration of nodes on the topology of the network are discussed in the present work this model predicts an observable two regime scale free degree distribution where the scaling exponents are 1 2 and 2 3 respectively the crossover point kcross of the degree distribution is adjusted by the growth rates of nodes and edges the nontrivial clustering coefficient and degree assortativity coefficient are relevant to the acceleration of nodes and edges

modeling the evolution of weighted networks we present a general model for the growth of weighted networks in which the structural growth is coupled with the edges weight dynamical evolution the model is based on a simple weight driven dynamics and a weights reinforcement mechanism coupled to the local network growth that coupling can be generalized in order to include the effect of additional randomness and nonlinearities which can be present in real world networks the model generates weighted graphs exhibiting the statistical properties observed in several real world systems in particular the model yields a nontrivial time evolution of vertices properties and scale free behavior with exponents depending on the microscopic parameters characterizing the coupling rules very interestingly the generated graphs spontaneously achieve a complex hierarchical architecture characterized by clustering and connectivity correlations varying as a function of the vertices degree

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

matrices with banded inverses inversion algorithms and factorization of gauss markov processes the paper considers the inversion of full matrices whose inverses are l banded we derive a nested inversion algorithm for such matrices applied to a tridiagonal matrix the algorithm provides its explicit inverse as an element wise product hadamard product of three matrices when related to gauss markov random processes gmrp this result provides a closed form factored expression for the covariance matrix of a first order gmrp this factored form leads to the interpretation of a first order gmrp as the product of three independent processes a forward independent increments process a backward independent increments process and a variance stationary process we explore the nonuniqueness of the factorization and design it so that the forward and backward factor processes have minimum energy we then consider the issue of approximating general nonstationary gaussian processes by gauss markov processes under two optimality criteria the kullback leibler distance and maximum entropy the problem reduces to approximating general covariances by covariance matrices whose inverses are banded our inversion result is an efficient algorithmic solution to this problem we evaluate the information loss between the original process and its gauss markov approximation

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

the optimal hard threshold for singular values is 4 sqrt 3 we consider recovery of low rank matrices from noisy data by hard thresholding of singular values in which empirical singular values below a threshold are set to 0 we study the asymptotic mean squared error amse in a framework where the matrix size is large compared with the rank of the matrix to be recovered and the signal to noise ratio of the low rank piece stays constant the amse optimal choice of hard threshold in the case of n by n matrix in white noise of level is simply 4 3 n 2 309 n when is known or simply 2 858 y med when is unknown where y med is the median empirical singular value for nonsquare m by n matrices with m n the thresholding coefficients 4 3 and 2 858 are replaced with different provided constants that depend on m n asymptotically this thresholding rule adapts to unknown rank and unknown noise level in an optimal manner it is always better than hard thresholding at any other value and is always better than ideal truncated singular value decomposition tsvd which truncates at the true rank of the low rank matrix we are trying to recover hard thresholding at the recommended value to recover an n by n matrix of rank r guarantees an amse at most 3 nr 2 in comparison the guarantees provided by tsvd optimally tuned singular value soft thresholding and the best guarantee achievable by any shrinkage of the data singular values are 5 nr 2 6 nr 2 and 2 nr 2 respectively the recommended value for hard threshold also offers among hard thresholds the best possible amse guarantees for recovering matrices with bounded nuclear norm empirical evidence suggests that performance improvement over tsvd and other popular shrinkage rules can be substantial for different noise distributions even in relatively small n

recent developments of the sinc numerical methods abstract this paper gives a survey of recent developments of the sinc numerical methods a variety of sinc numerical methods have been developed by stenger and his school for a certain class of problems the sinc numerical methods have the convergence rates of o exp n with some 0 where n is the number of nodes or bases used in the methods recently it has turned out that the sinc numerical methods can achieve convergence rates of o exp n log n with some 0 for a smaller but still practically meaningful class of problems and that these convergence rates are best possible the present paper demonstrates these facts for two sinc numerical methods the sinc approximation and the sinc collocation method for two point boundary value problems

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

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

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

fast algorithms for high order spline collocation systems in this paper we present a complete eigenvalue analysis for arbitrary order c 1 spline collocation methods applied to the poisson equation on a rectangular domain with dirichlet boundary conditions based on this analysis we develop some fast algorithms for solving a class of high order spline collocation systems which arise from discretizing the poisson equation

numerical methods based on whittaker cardinal or sinc functions this paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis described in this paper are methods of interpolation and approximation of functions and their derivatives quadrature the approximate evaluation of transforms hilbert fourier laplace hankel and mellin and the approximate solution of differential and integral equations the methods have many advantages over classical methods which use polynomials as bases in addition all of the methods converge at an optimal rate if singularities on the boundary of approximation are ignored

a fast poisson solver for the finite difference solution of the incompressible navier stokes equations in this paper a fast direct solver for the poisson equation on the half staggered grid is presented the poisson equation results from the projection method of the finite difference solution of the incompressible navier stokes equations to achieve our goal new algorithms for diagonalizing a semidefinite pair are developed the fast solver can also be extended to the three dimensional case the motivation and related issues in using this half staggered grid are also discussed

robust shrinkage estimation of high dimensional covariance matrices we address high dimensional covariance estimation for elliptical distributed samples which are also known as spherically invariant random vectors sirv or compound gaussian processes specifically we consider shrinkage methods that are suitable for high dimensional problems with a small number of samples large p small n we start from a classical robust covariance estimator tyler 1987 which is distribution free within the family of elliptical distribution but inapplicable when n p using a shrinkage coefficient we regularize tyler s fixed point iterations we prove that for all n and p the proposed fixed point iterations converge to a unique limit regardless of the initial condition next we propose a simple closed form and data dependent choice for the shrinkage coefficient which is based on a minimum mean squared error framework simulations demonstrate that the proposed method achieves low estimation error and is robust to heavy tailed samples finally as a real world application we demonstrate the performance of the proposed technique in the context of activity intrusion detection using a wireless sensor network

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

network properties of written human language we investigate the nature of written human language within the framework of complex network theory in particular we analyze the topology of orwell s 1984 focusing on the local properties of the network such as the properties of the nearest neighbors and the clustering coefficient we find a composite power law behavior for both the average nearest neighbor s degree and average clustering coefficient as a function of the vertex degree this implies the existence of different functional classes of vertices furthermore we find that the second order vertex correlations are an essential component of the network architecture to model our empirical results we extend a previously introduced model for language due to dorogovtsev and mendes we propose an accelerated growing network model that contains three growth mechanisms linear preferential attachment local preferential attachment and the random growth of a predetermined small finite subset of initial vertices we find that with these elementary stochastic rules we are able to produce a network showing syntacticlike structures

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

double exponential transformation in the sinc collocation method for two point boundary value problems in 1974 takahasi and mori proposed the double exponential transformation for efficient evaluation of integrals of an analytic function with end point singularity recently it has turned out that the double exponential transformation is also useful for various kinds of sinc methods in the present paper we consider the sinc collocation method for two point boundary value problems to show that the method can be considerably enhanced when incorporated with the double exponential transformation technique

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved