relative entropy and the multivariable multidimensional moment problem entropy like functionals on operator algebras have been studied since the pioneering work of von neumann umegaki lindblad and lieb the best known are the von neumann entropy rho trace rhologrho and a generalization of the kullback leibler distance s rhoparsigma trace rhologrho rhologsigma referred to as quantum relative entropy and used to quantify distance between states of a quantum system the purpose of this paper is to explore and s as regularizing functionals in seeking solutions to multivariable and multidimensional moment problems it will be shown that extrema can be effectively constructed via a suitable homotopy the homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems this is accomplished by a renormalization of a riemannian metric induced by entropy functionals as an application we discuss the inverse problem of describing power spectra which are consistent with second order statistics which has been the main motivation behind the present work

on the estimation of structured covariance matrices this paper discusses a method for estimating the covariance matrix of a multivariate stationary process w generated as the output of a given linear filter fed by a stationary process y the estimated covariance matrix must satisfy two constraints it must be positive semi definite and it must be consistent with the fact that w is the output of the given linear filter it turns out that these constraints force the estimated covariance to lie in the intersection of a cone with a linear space while imposing only the first of the two constraints is rather straightforward guaranteeing that both are satisfied is a non trivial issue to which quite a bit of attention has already been devoted in the literature our approach extends the method for estimating the toeplitz covariance matrix of order m of a process y based on the biased spectral estimator stoica moses 1997 this extension is based on the characterization of the output covariance matrix in terms of the filter parameters and the sequence of covariance lags of the input process after introducing our estimation method we propose a comparison performance between this one and other methods proposed in the literature simulation results show that our approach constitutes a valid estimation procedure

distances and riemannian metrics for multivariate spectral densities the paper is concerned with the practical problem of how to compare power spectral densities of multivariable time series for scalar time series several notions of distance divergences have been proposed and studied starting from the early 1970s while multivariable ones have only recently began to receive any attention in the paper two classes of divergence measures inspired by classical prediction theory are introduced these divergences naturally induce riemannian metrics on the cone of multivariable densities the metrics amount to the quadratic term in the divergence between infinitesimally close to each other power spectra for one of the two we provide explicit formulae for the corresponding geodesics and geodesic distance a close connection between the geometry of power spectra and the geometry of the fisher rao metric is noted

the exchange property for row and column finite matrix rings abstract the ring b r of all row and column finite matrices over any unital von neumann regular ring r is shown to be an exchange ring more generally the same is true of the multiplier ring of a non unital regular ring with a countable unit

strongly clean matrix rings over local rings an element of a ring r with identity is called strongly clean if it is the sum of an idempotent and a unit that commute and r is called strongly clean if every element of r is strongly clean in this paper we determine when a 2 2 matrix a over a commutative local ring is strongly clean several equivalent criteria are given for such a matrix to be strongly clean consequently we obtain several equivalent conditions for the 2 2 matrix ring over a commutative local ring to be strongly clean extending a result of chen yang and zhou

on the robustness of the bayes and wiener estimators under model uncertainty abstract this paper deals with the robust estimation problem of a signal given noisy observations we assume that the actual statistics of the signal and observations belong to a ball about the nominal statistics this ball is formed by placing a bound on a suitable divergence or distance between the actual and the nominal statistics then the robust estimator is obtained by minimizing the mean square error according to the least favorable statistics in that ball therefore we obtain a divergence based minimax approach to robust estimation choosing a set of divergences called tau divergence family we show that the bayes estimator based on the nominal statistics is the optimal solution moreover in the dynamic case the optimal offline estimator is the noncausal wiener filter based on the nominal statistics

condition number related with w weighted drazin inverse and singular linear systems in this paper we introduce a pq norm and discuss the condition number of the w weighted drazin inverse we get some results on the condition number with respect to a d w and singular linear systems wawx b b r wa k 2 x r aw k 1 which are similar to the regular inverse a 1 and nonsingular linear systems ax b

when is the 2 2 matrix ring over a commutative local ring strongly clean a ring r with identity is called strongly clean if every element of r is the sum of an idempotent and a unit that commute local rings are strongly clean it is unknown when a matrix ring is strongly clean however it is known from j chen x yang y zhou on strongly clean matrix and triangular matrix rings preprint 2005 that for any prime number p the 2 2 matrix ring m2 z p is strongly clean where z p is the ring of p adic integers but m2 z p is not strongly clean where z p is the localization of z at the prime ideal generated by p let r be a commutative local ring a criterion in terms of solvability of a simple quadratic equation in r is obtained for m2 r to be strongly clean as consequences m2 r is strongly clean iff m2 r x is strongly clean iff m2 r x xn is strongly clean iff m2 rc2 is strongly clean

some families of strongly clean rings abstract a ring r with identity is called strongly clean if every element of r is the sum of an idempotent and a unit that commute with each other for a commutative local ring r and for an arbitrary integer n 2 the paper deals with the question whether the strongly clean property of m n r x m n r x x k and m n rc 2 follows from the strongly clean property of m n r this is yes if n 2 by a known result

a new approach to spectral estimation a tunable high resolution spectral estimator traditional maximum entropy spectral estimation determines a power spectrum from covariance estimates here we present a new approach to spectral estimation which is based on the use of filter banks as a means of obtaining spectral interpolation data such data replaces standard covariance estimates a computational procedure for obtaining suitable pole zero arma models from such data is presented the choice of the zeros ma part of the model is completely arbitrary by suitable choices of filter bank poles and spectral zeros the estimator can be tuned to exhibit high resolution in targeted regions of the spectrum

robust and efficient estimation by minimising a density power divergence a minimum divergence estimation method is developed for robust parameter estimation the proposed approach uses new density based divergences which unlike existing methods of this type such as minimum hellinger distance estimation avoid the use of nonparametric density estimation and associated complications such as bandwidth selection the proposed class of density power divergences is indexed by a single parameter which controls the trade off between robustness and efficiency the methodology affords a robust extension of maximum likelihood estimation for which 0 choices of near zero afford considerable robustness while retaining efficiency close to that of maximum likelihood

strongly clean matrices over arbitrary rings abstract we characterize when the companion matrix of a monic polynomial over an arbitrary ring r is strongly clean in terms of a type of ideal theoretic factorization which we call an isrc factorization in the polynomial ring r t this provides a nontrivial necessary condition for m n r to be strongly clean for r arbitrary if the ring in question is either local or commutative then we can say more generalizing and extending most of what is currently known about this problem if r is local our isrc factorization is equivalent to an actual polynomial factorization generalizing results in 1 18 and 12 if instead r is commutative and h r t is monic we again show that an isrc factorization yields a polynomial factorization and we prove that h has such a factorization if and only if its companion matrix is strongly clean if and only if every algebraic element in every r algebra which satisfies h is strongly clean this generalizes the work done in 1 on commutative local rings and provides a characterization of strong cleanness in m n r for any commutative ring r

a globally convergent matricial algorithm for multivariate spectral estimation in this paper we first describe a matricial newton type algorithm designed to solve the multivariable spectrum approximation problem we then prove its global convergence finally we apply this approximation procedure to multivariate spectral estimation and test its effectiveness through simulation simulation shows that in the case of short observation records this method may provide a valid alternative to standard multivariable identification techniques such as matlab s pem and matlab s n4sid

multivariate spectral estimation based on the concept of optimal prediction in this technical note we deal with a spectrum approximation problem arising in three like multivariate spectral estimation approaches the solution to the problem minimizes a suitable divergence index with respect to an a priori spectral density we derive a new divergence family between multivariate spectral densities which takes root in the prediction theory under mild assumptions on the a priori spectral density the approximation problem based on this new divergence family admits a family of solutions moreover an upper bound on the complexity degree of these solutions is provided

a new family of high resolution multivariate spectral estimators in this paper we extend the beta divergence family to multivariate power spectral densities similarly to the scalar case we show that it smoothly connects the multivariate kullback leibler divergence with the multivariate itakura saito distance we successively study a spectrum approximation problem based on the beta divergence family which is related to a multivariate extension of the three spectral estimation technique it is then possible to characterize a family of solutions to the problem an upper bound on the complexity of these solutions will also be provided finally we will show that the most suitable solution of this family depends on the specific features required from the estimation problem

strong cleanness of the 2 2 matrix ring over a general local ring abstract a ring r is called strongly clean if every element of r is the sum of a unit and an idempotent that commute with each other a recent result of borooah diesl and dorsey g borooah a j diesl t j dorsey strongly clean matrix rings over commutative local rings j pure appl algebra 212 1 2008 281 296 completely characterized the commutative local rings r for which m n r is strongly clean for a general local ring r and n 1 however it is unknown when the matrix ring m n r is strongly clean here we completely characterize the local rings r for which m 2 r is strongly clean

an interpretation of the dual problem of the three like approaches spectral estimation can be performed using the so called three like approach such method leads to a convex optimization problem whose solution is characterized through its dual problem in this paper we show that the dual problem can be seen as a new parametric spectral estimation problem this interpretation implies that the three like solution is optimal in terms of closeness to the correlogram over a certain parametric class of spectral densities describing arma models enriching in this way its meaningfulness

clean matrices and unit regular matrices an element in a ring r is said to be clean respectively unit regular if it is the sum respectively product of an idempotent element and an invertible element if all elements in r are unit regular it is known that all elements in r are clean in this note we show that a single unit regular element in a ring need not be clean more generally a criterion is given for a matrix ab00 to be clean in a matrix ring m2 k over any commutative ring k for k z this criterion shows for instance that the unit regular matrix 12500 is not clean also this turns out to be the smallest such example

the structure of state covariances and its relation to the power spectrum of the input we study the relationship between power spectra of stationary stochastic inputs to a linear filter and the corresponding state covariances and identify the structure of positive semidefinite matrices that qualify as state covariances of the filter this structure is best revealed by a rank condition pertaining to the solvability of a linear equation involving the state covariance and the system matrices we then characterize all input power spectra consistent with any specific state covariance the parametrization of input spectra is achieved through a relation to solutions of an analytic interpolation problem which is analogous but not equivalent to a matricial nehari problem

strong cleanness of generalized matrix rings over a local ring abstract let k s r be the generalized matrix ring over a ring r with multiplier s for a general local ring r and a central element s in the jacobson radical of r necessary and sufficient conditions are obtained for k s r to be a strongly clean ring for a commutative local ring r and an arbitrary element s in r criteria are obtained for a single element of k s r to be strongly clean and respectively for the ring k s r to be strongly clean specializing to s 1 yields some known results new families of strongly clean rings are presented

a maximum entropy enhancement for a family of high resolution spectral estimators structured covariances occurring in spectral analysis filtering and identification need to be estimated from a finite observation record the corresponding sample covariance usually fails to possess the required structure this is the case for instance in the byrnes georgiou lindquist three like tunable high resolution spectral estimators there the output covariance of a linear filter is needed to initialize the spectral estimation technique the sample covariance estimate however is usually not compatible with the filter in this paper we present a new systematic way to overcome this difficulty the new estimate is obtained by solving an ancillary problem with an entropic type criterion extensive scalar and multivariate simulation shows that this new approach consistently leads to a significant improvement of the spectral estimators performances

further results on the convergence of the pavon ferrante algorithm for spectral estimation in this paper we provide a detailed analysis of the global convergence properties of an extensively studied and extremely effective fixed point algorithm for the kullback leibler approximation of spectral densities proposed by pavon and ferrante in their paper on the georgiou lindquist approach to constrained kullback leibler approximation of spectral densities our main result states that the algorithm globally converges to one of its fixed points

time and spectral domain relative entropy a new approach to multivariate spectral estimation the concept of spectral relative entropy rate is introduced for jointly stationary gaussian processes using classical information theoretic results we establish a remarkable connection between time and spectral domain relative entropy rates this naturally leads to a new spectral estimation technique where a multivariate version of the itakura saito distance is employed it may be viewed as an extension of the approach called three introduced by byrnes georgiou and lindquist in 2000 which in turn followed in the footsteps of the burg jaynes maximum entropy method spectral estimation is here recast in the form of a constrained spectrum approximation problem where the distance is equal to the processes relative entropy rate the corresponding solution entails a complexity upper bound which improves on the one so far available in the multichannel framework indeed it is equal to the one featured by three in the scalar case the solution is computed via a globally convergent matricial newton type algorithm simulations suggest the effectiveness of the new technique in tackling multivariate spectral estimation tasks especially in the case of short data records

robust kalman filtering under model perturbations we consider a family of divergence based minimax approaches to perform robust filtering the mismodeling budget or tolerance is specified at each time increment of the model more precisely all possible model increments belong to a ball which is formed by placing a bound on the tau divergence family between the actual and the nominal model increment then the robust filter is obtained by minimizing the mean square error according to the least favorable model in that ball it turns out that the solution is a family of kalman like filters their gain matrix is updated according to a risk sensitive like iteration where the risk sensitivity parameter is now time varying as a consequence we also extend the risk sensitive filter to a family of risk sensitive like filters according to the tau divergence family