maximum likelihood estimation of the autoregressive model by relaxation on the reflection coefficients a method for autoregressive parameter estimation which successively maximizes the likelihood with respect to each reflection coefficient while keeping the others fixed is presented the algorithm generalizes the recursive maximum likelihood technique of s m kay 1983 which corresponds to performing only one iteration cycle an interesting application is the estimation of a toeplitz covariance matrix simulations show that the algorithm converges quite fast and provides much better estimates than current procedures for short record length

on the identification of certain rational transfer functions from truncated autocovariance sequences let h z spl sigma sub k 0 sup spl infin h sub k z sup k be a q spl times p rational causal minimum phase transfer function and let r sub n sub n spl isin z be the autocovariance sequence associated to the spectral density h z h sup t z sup 1 in this paper the authors address the identification problem of h z from the truncated autocovariance sequence r sub n sub n spl ne 0 for the case q p

on periodic autoregressive processes estimation we consider the autoregressive estimation for periodically correlated processes using the parametrization given by the partial autocorrelation function we propose an estimation of these parameters by extending the sample partial autocorrelation method to this situation a comparison with other methods is made relationships with the stationary multivariate case are discussed

mathematical considerations for the problem of fourier transform phase retrieval from magnitude in this paper we deal with the problem of retrieving a finite extent function from the magnitude of its fourier transform this so called phase retrieval problem will first be posed under its different underlying models we will present a brief review of the main results known in this area for both discrete and continuous phase retrieval models giving special emphasis to the algebraic problem of the uniqueness of the solution several important issues which are yet unresolved will be pointed out and discussed we will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real valued sequence x from the magnitude of the output of a linear distortion hx j j 1 cdots n a number of important results will be obtained for this general setting by means of algebraic geometric techniques in particular the problems of the existence of a solution for phase retrieval number of feasible solutions stability of the essential

the reconstruction of a multidimensional sequence from the phase or magnitude of its fourier transform this paper addresses two fundamental issues involved in the reconstruction of a multidimensional sequence from either the phase or magnitude of its fourier transform the first issue relates to the uniqueness of a multidimensional sequence in terms of its phase or magnitude although phase or magnitude information alone is not sufficient in general to uniquely specify a sequence a large class of sequences are shown to be recoverable from their phase or magnitude the second issue which is addressed in this paper concerns the actual reconstruction of a multidimensional sequence from its phase or magnitude for those sequences which are uniquely specified by their phase several practical algorithms are described which may be used to reconstruct a sequence from its phase several examples of phase only reconstruction are also presented unfortunately however even for those sequences which are uniquely defined by their magnitude it appears that a practical algorithm is yet to be developed for reconstructing a sequence from only its magnitude nevertheless an iterative procedure which has been proposed is briefly discussed and evaluated

recursive maximum likelihood estimation of autoregressive processes a new method of autoregressive parameter estimation is presented the technique is a closer approximation to the true maximum likelihood estimator than that obtained using linear prediction techniques the advantage of the new algorithm is mainly for short data records and or sharply peaked spectra simulation results indicate that the parameter bias as well as the variance is reduced over the yule walker and the forward backward approaches of linear prediction also spectral estimates exhibit more resolution and less spurious peaks a stable all pole filter estimate is guaranteed the algorithm operates in a recursive model order fashion which allows one to successively fit higher order models to the data

a necessary and sufficient condition for the existence of the maximum likelihood estimate in autoregressive models the author studies the existence of a maximum likelihood estimate mle for the parameters of a pth order autoregressive ar model from n or 1 independent records of length m of a complex time series it is shown that for almost all such set of observations the mle exists if and only if the n records be exactly fitted by complex undamped sinusoids using the same set of p distinct frequencies

a riemannian symmetric rank one trust region method the well known symmetric rank one trust region method where the hessian approximation is generated by the symmetric rank one update is generalized to the problem of minimizing a real valued function over a d d dimensional riemannian manifold the generalization relies on basic differential geometric concepts such as tangent spaces riemannian metrics and the riemannian gradient as well as on the more recent notions of first order retraction and vector transport the new method called rtr sr1 is shown to converge globally and d 1 d 1 step q superlinearly to stationary points of the objective function a limited memory version referred to as lrtr sr1 is also introduced in this context novel efficient strategies are presented to construct a vector transport on a submanifold of a euclidean space numerical experiments rayleigh quotient minimization on the sphere and a joint diagonalization problem on the stiefel manifold illustrate the value of the new methods

identification of certain noisy ma models new results abstract in this paper we address the identification problem of p inputs q outputs ma models corrupted by a white noise with unknown covariance matrix in the case where p q under certain additional conditions we show that the generating function of the ma model is identifiable up to a p p constant orthogonal matrix from the autocovariance function of the observation our results extend those already obtained in desbouvries et al 5 and desbouvries and loubaton 6

optimization methods on riemannian manifolds and their application to shape space we extend the scope of analysis for linesearch optimization algorithms on possibly infinite dimensional riemannian manifolds to the convergence analysis of the bfgs quasi newton scheme and the fletcher reeves conjugate gradient iteration numerical implementations for exemplary problems in shape spaces show the practical applicability of these methods

a subspace algorithm for certain blind identification problems the problem of blind identification of p inputs q outputs fir transfer functions is addressed existing subspace identification methods derived for p 1 are first reformulated in particular the links between the noise subspace of a certain covariance matrix of the output signals on which subspace methods build on and certain rational subspaces associated with the transfer function to be identified are elucidated based on these relations we study the behavior of the subspace method in the case where the order of the transfer function is overestimated next an asymptotic performance analysis of this estimation method is carried out consistency and asymptotical normality of the estimates is established a closed form expression for the asymptotic covariance of the estimates is given numerical simulations and investigations are presented to demonstrate the potential of the subspace method finally we take advantage of our new reformulation to discuss the extension of the subspace method to the case p 1 we show where the difficulties lie and we briefly indicate how to solve the corresponding problems the possible connections with classical approaches for ma model estimations are also outlined

identification of rank one rational spectral densities from noisy observations a stochastic realization approach abstract let s be a q q spectral density matrix which is the sum of a rational rank one spectral density s 1 and of a constant positive definite matrix q the identification problem of s 1 and q from s is addressed the conditions under which s 1 and q are identifiable are first derived then an identification method is proposed it is based on a parametrization of the external stochastic realizations of s whose innovation sequence has a prescribed dimension

second order blind separation of sources based on canonical partial innovations this paper is devoted to the study of the second order properties using partial autocorrelations of an instantaneous mixture of colored sources without additive noise we introduce the notion of symmetric recursive canonical partial innovation then their components for the observation process meet exactly with those of the source process from the order for which the autoregressive models underlying the sources are distinct this property leads to a new separation method based on the sample counterpart of partial autocorrelation matrices associated with these innovations simulation results show a notable improvement of the achievements of such an approach with respect to those of similar methods two other separation methods related to partial autocorrelation are also discussed

on the identification of noisy ma models in this paper we address the identification problem of p input q output ma models corrupted by a white noise with an unknown covariance matrix in the case where p q under certain additional conditions we show that the generating function of the ma model is identifiable from the autocovariance function of the observation some simple algebraic identification procedures are also given

on local maxima of the likelihood function for toeplitz matrix estimation it is shown that in estimating a toeplitz covariance matrix the likelihood function can present a local maximum close to the global maximum an analytical proof is given for the matrices of order 3 and numerical examples are shown for orders 3 4 and 5

solving phaselift by low rank riemannian optimization methods for complex semidefinite constraints a framework phaselift was recently proposed to solve the phase retrieval problem in this framework the problem is solved by optimizing a cost function over the set of complex hermitian positive semidefinite matrices this approach to phase retrieval motivates a more general consideration of optimizing cost functions on semidefinite hermitian matrices where the desired minimizers are known to have low rank this paper considers an approach based on an alternative cost function defined on a union of appropriate manifolds it is related to the original cost function in a manner that preserves the ability to find a global minimizer and is significantly more efficient computationally a rank based optimality condition for stationary points is given and optimization algorithms based on state of the art riemannian optimization and dynamically reducing rank are proposed empirical evaluations are performed using the phaselift problem the new approach is shown to be an effective method of phase retrieval with

properties and applications of gaussian autoregressive processes in detection theory corresp a sufficient statistic having dimension 3p 1 is constructed for p th order stationary gaussian autoregressive processes a computationally efficient discriminator based on the statistic is obtained a derivation of the whitening filter correlator detector for known signals in autoregressive noise is presented a new formula for the signal to noise ratio of an optimal detector for a constant signal in stationary correlated gaussian noise is presented and used to help study the nature of autoregressive approximations to more general processes in this application

descent algorithms on oblique manifold for source adaptive ica contrast a riemannian manifold optimization strategy is proposed to facilitate the relaxation of the orthonormality constraint in a more natural way in the course of performing independent component analysis ica that employs a mutual information based source adaptive contrast function despite the extensive development of manifold techniques catering to the orthonormality constraint only a limited number of works have been dedicated to oblique manifold ob algorithms to intrinsically handle the normality constraint which has been empirically shown to be superior to other riemannian and euclidean approaches imposing the normality constraint implicitly in line with the ica definition essentially guarantees a substantial improvement in the solution accuracy by way of increased degrees of freedom while searching for an optimal unmixing ica matrix in contrast with the orthonormality constraint designs of the steepest descent conjugate gradient with hager zhang or a hybrid update parameter quasi newton and cost effective quasi newton methods intended for ob are presented in this paper their performance is validated using natural images and systematically compared with the popular state of the art approaches in order to assess the performance effects of the choice of algorithm and the use of a riemannian rather than euclidean framework we surmount the computational challenge associated with the direct estimation of the source densities using the improved fast gauss transform in the evaluation of the contrast function and its gradient the proposed ob schemes may find applications in the offline image signal analysis wherein on one hand the computational overhead can be tolerated and on the other the solution quality holds paramount interest

intrinsic representation of tangent vectors and vector transports on matrix manifolds the quasi newton methods on riemannian manifolds proposed thus far do not appear to lend themselves to satisfactory convergence analyses unless they resort to an isometric vector transport this prompts us to propose a computationally tractable isometric vector transport on the stiefel manifold of orthonormal p frames in mathbb r n rn specifically it requires o np 2 o np2 flops which is considerably less expensive than existing alternatives in the frequently encountered case where n gg p n p we then build on this result to also propose computationally tractable isometric vector transports on other manifolds namely the grassmann manifold the fixed rank manifold and the positive semidefinite fixed rank manifold in the process we also propose a convenient way to represent tangent vectors to these manifolds as elements of mathbb r d rd where d is the dimension of the manifold we call this an intrinsic representation as opposed to extrinsic representations as elements of mathbb r w rw where w is the dimension of the embedding space finally we demonstrate the performance of the proposed isometric vector transport in the context of a riemannian quasi newton method applied to minimizing the brockett cost function

efficient computation of autoregressive estimates through a sufficient statistic it is shown that various time reversible methods in particular burg s algorithm for autoregressive model estimation may be performed through the use of a simple sufficient statistic this provides more efficient computation of the estimators

phaselift exact and stable signal recovery from magnitude measurements via convex programming suppose we wish to recover a signal input amssym font abc cmmib10 def bi 1 hbox abc 1 bi x in bbb c n from m intensity measurements of the form that is from data in which phase information is missing we prove that if the vectors are sampled independently and uniformly at random on the unit sphere then the signal x can be recovered exactly up to a global phase factor by solving a convenient semidefinite program a trace norm minimization problem this holds with large probability provided that m is on the order of and without any assumption about the signal whatsoever this novel result demonstrates that in some instances the combinatorial phase retrieval problem can be solved by convex programming techniques finally we also prove that our methodology is robust vis a vis additive noise 2012 wiley periodicals inc