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1 On the Covariance Completion Problem under a Circulant Structure
"... Abstract — Covariance matrices with a circulant structure arise in the context of discretetime periodic processes and their significance stems also partly from the fact that they can be diagonalized via a Fourier transformation. This note deals with the problem of completion of partially specified ..."
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Abstract — Covariance matrices with a circulant structure arise in the context of discretetime periodic processes and their significance stems also partly from the fact that they can be diagonalized via a Fourier transformation. This note deals with the problem of completion of partially specified circulant covariance matrices. The particular completion that has maximal determinant (i.e., the socalled maximum entropy completion) was considered in Carli etal. [2] where it was shown that if a single band is unspecified and to be completed, the algebraic restriction that enforces the circulant structure is automatically satisfied and that the inverse of the maximizer has a band of zero values that corresponds to the unspecified band in the data—i.e., it has the Dempster property. The purpose of the present note is to develop an independent proof of this result which in fact extends naturally to any number of missing bands as well as arbitrary missing elements. More specifically, we show that this general fact is a direct consequence of the invariance of the determinant under the group of transformations that leave circulant matrices invariant. A description of the complete set of all positive extensions of partially specified circulant matrices is also given and certain connections between such sets and the factorization of certain polynomials in many variables, facilitated by the circulant structure, is highlighted. I.
Matrix convex functions with applications to weighted centres for semidefinite programming
, 2005
"... In this paper, we develop various calculus rules for general smooth matrixvalued functions and for the class of matrix convex (or concave) functions first introduced by Löwner and Kraus in 1930s. Then we use these calculus rules and the matrix convex function − logX to study a new notion of weight ..."
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In this paper, we develop various calculus rules for general smooth matrixvalued functions and for the class of matrix convex (or concave) functions first introduced by Löwner and Kraus in 1930s. Then we use these calculus rules and the matrix convex function − logX to study a new notion of weighted centers for semidefinite programming (SDP) and show that, with this definition, some known properties of weighted centers for linear programming can be extended to SDP. We also show how the calculus rules for matrix convex functions can be used in the implementation of barrier methods for optimization problems involving nonlinear matrix functions.
1 Geometric Methods for Spectral Analysis
"... Abstract—This paper explores a geometric framework for modeling nonstationary but slowly varying time series, based on the assumption that shortwindowed power spectra capture their spectral character, and that energy transference in the frequency domain has a physical significance. The framework r ..."
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Abstract—This paper explores a geometric framework for modeling nonstationary but slowly varying time series, based on the assumption that shortwindowed power spectra capture their spectral character, and that energy transference in the frequency domain has a physical significance. The framework relies on certain notions of transportation distance and their respective geodesics to model possible nonparametric changes in the power spectral density with respect to time. We discuss the relevance of this framework to applications in spectral tracking, spectral averaging, and speech morphing. Index Terms—spectral metrics, geodesics, transportation distance, spectral analysis, spectral tracking, spectral averaging, speech morphing I.
Uncertainty bounds for spectral estimation
 IEEE Transactions on Automatic Control
, 2013
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Semidefinite programming for gradient and Hessian computation in maximum entropy estimation
 Proceedings 48th IEEE CDC Conference, NewOrleans (2007
"... Abstract — We consider the classical problem of estimating a density on [0,1] via some maximum entropy criterion. For solving this convex optimization problem with algorithms using firstorder or secondorder methods, at each iteration one has to compute (or at least approximate) moments of some mea ..."
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Abstract — We consider the classical problem of estimating a density on [0,1] via some maximum entropy criterion. For solving this convex optimization problem with algorithms using firstorder or secondorder methods, at each iteration one has to compute (or at least approximate) moments of some measure with a density on [0,1], to obtain gradient and Hessian data. We propose a numerical scheme based on semidefinite programming that avoids computing quadrature formula for this gradient and Hessian computation. I.
On the Geometry of Maximum Entropy Problems
 SIAM Review. A Publication of the Society for Industrial and Applied Mathematics
"... Abstract. We show that a simple geometric result suffices to derive the form of the optimal solution in a large class of finite and infinitedimensional maximum entropy problems concerning probability distributions, spectral densities, and covariance matrices. These include Burg’s spectral estimati ..."
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Abstract. We show that a simple geometric result suffices to derive the form of the optimal solution in a large class of finite and infinitedimensional maximum entropy problems concerning probability distributions, spectral densities, and covariance matrices. These include Burg’s spectral estimation method and Dempster’s covariance completion, as well as various recent generalizations of the above. We then apply this orthogonality principle to the new problem of completing a blockcirculant covariance matrix when an a priori estimate is available.
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"... This thesis is concerned with two generalized moment problems arising in the estimation of stochastic models. Firstly, we consider the THREE approach, introduced by Byrnes Georgiou and Lindquist, for estimating spectral densities. Here, the output covariance matrix of a known bank of filters is us ..."
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This thesis is concerned with two generalized moment problems arising in the estimation of stochastic models. Firstly, we consider the THREE approach, introduced by Byrnes Georgiou and Lindquist, for estimating spectral densities. Here, the output covariance matrix of a known bank of filters is used to extract information on the input spectral density which needs to be estimated. The parametrization of the family of spectral densities matching the output covariance is a generalized moment problem. An estimate of the input spectral density is then chosen from this family. The choice criterium is based on the minimization of a suitable divergence index among spectral densities. After the introduction of the THREElike paradigm, we present a multivariate extension of the Beta divergence for solving the problem. Afterward, we deal with the estimation of the output covariance of the filters bank given a finitelength data generated by the unknown input spectral density. Secondly, we deal with the quantum process tomography. This problem consists in the estimation of a quantum channel which can be thought as the quantum equivalent of the Markov transition matrix in the classical setting. Here, a quantum system prepared in a known pure state is fed to the unknown channel. A measurement of an observable is performed on the output state. The set of the employed pure states and observables represents the experimental setting. Again, the parametrization of the family of quantum channels matching the measurements is a generalized moment problem. The choice criterium for the best estimate in this family is based on the maximization of maximum likelihood functionals. The corresponding estimate, however, may not be unique since the experimental setting is not “rich” enough in many cases of interest. We characterize the minimal experimental setting which guarantees the uniqueness of the estimate. Numerical simulation evidences that experimental settings richer than the minimal one do not lead to better performances. i