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54
Relative Entropy and the multivariable multidimensional Moment Problem
 IEEE Trans. on Information Theory
"... Entropylike functionals on operator algebras have been studied since the pioneering work of von Neumann, Umegaki, Lindblad, and Lieb. The most wellknown are the von Neumann entropy I(ρ): = −trace(ρ log ρ) and a generalization of the KullbackLeibler distance S(ρσ): = trace(ρ log ρ − ρ log σ), re ..."
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Cited by 29 (7 self)
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Entropylike functionals on operator algebras have been studied since the pioneering work of von Neumann, Umegaki, Lindblad, and Lieb. The most wellknown are the von Neumann entropy I(ρ): = −trace(ρ log ρ) and a generalization of the KullbackLeibler distance S(ρσ): = trace(ρ log ρ − ρ log σ), refered to as quantum relative entropy and used to quantify distance between states of a quantum system. The purpose of this paper is to explore I 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 secondorder statistics, which has been the main motivation behind the present work.
Generalized interpolation in H∞ with a complexity constraint
 TRANS. AMER. MATH. SOC
, 2006
"... In a seminal paper, Sarason generalized some classical interpolation problems for H∞ functions on the unit disc to problems concerning lifting onto H² of an operator T that is defined on K = H² ⊖φH² (φ is an inner function) and commutes with the (compressed) shift S. In particular, he showed that in ..."
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Cited by 27 (13 self)
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In a seminal paper, Sarason generalized some classical interpolation problems for H∞ functions on the unit disc to problems concerning lifting onto H² of an operator T that is defined on K = H² ⊖φH² (φ is an inner function) and commutes with the (compressed) shift S. In particular, he showed that interpolants (i.e., f ∈ H ∞ such that f(S)=T) having norm equal to �T � exist, and that in certain cases such an f is unique and can be expressed as a fraction f = b/a with a, b ∈ K. In this paper, we study interpolants that are such fractions of K functions and are bounded in norm by 1 (assuming that �T � < 1, in which case they always exist). We parameterize the collection of all such pairs (a, b) ∈ K × K and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where φ is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.
A maximum entropy enhancement for a family of highresolution spectral estimators
 IEEE Trans. Autom. Control
, 2012
"... Abstract—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 T ..."
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Cited by 14 (6 self)
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Abstract—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 THREElike tunable, highresolution 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 entropictype criterion. Extensive scalar and multivariate simulation shows that this new approach consistently leads to a significant improvement of the spectral estimators performances. Index Terms—Convex optimization, covariance extension, maximum entropy, multivariable spectral estimation. I.
A Convex Optimization Approach to ARMA Modeling
"... Abstract—We formulate a convex optimization problem for approximating any given spectral density with a rational one having a prescribed number of poles and zeros (n poles and m zeros inside the unit disc and their conjugates). The approximation utilizes the Kullback–Leibler divergence as a distance ..."
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Cited by 10 (1 self)
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Abstract—We formulate a convex optimization problem for approximating any given spectral density with a rational one having a prescribed number of poles and zeros (n poles and m zeros inside the unit disc and their conjugates). The approximation utilizes the Kullback–Leibler divergence as a distance measure. The stationarity condition for optimality requires that the approximant matches n +1covariance moments of the given power spectrum and m cepstral moments of the corresponding logarithm, although the latter with possible slack. The solution coincides with one derived by Byrnes, Enqvist, and Lindquist who addressed directly the question of covariance and cepstral matching. Thus, the present paper provides an approximation theoretic justification of such a problem. Since the approximation requires only moments of spectral densities and of their logarithms, it can also be used for system identification. Index Terms—ARMA modeling, cepstral coefficients, convex optimization, covariance matching. I.
The Inverse Problem of Analytic Interpolation with Degree Constraint and Weight Selection for Control Synthesis
"... The minimizers of certain weighted entropy functionals are the solutions to an analytic interpolation problem with a degree constraint, and all solutions to this interpolation problem arise in this way by a suitable choice of weights. Selecting appropriate weights is pertinent to feedback control sy ..."
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Cited by 9 (7 self)
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The minimizers of certain weighted entropy functionals are the solutions to an analytic interpolation problem with a degree constraint, and all solutions to this interpolation problem arise in this way by a suitable choice of weights. Selecting appropriate weights is pertinent to feedback control synthesis, where interpolants represent closedloop transfer functions. In this paper we consider the correspondence between weights and interpolants in order to systematize feedback control synthesis with a constraint on the degree. There are two basic issues that we address: we first characterize admissible shapes of minimizers by studying the corresponding inverse problem, and then we develop effective ways of shaping minimizers via suitable choices of weights.
The moment problem for rational measures: convexity in the spirit of Krein
 in Modern Analysis and Application: To the centenary of Mark Krein, Vol. I: Operator Theory and Related Topics, Birkhäuser
, 2009
"... In memory of Mark Grigoryevich Krein on the occasion of the 100th anniversary of his birth Abstract. The moment problem as formulated by Krein and Nudel’man is a beautiful generalization of several important classical moment problems, including the power moment problem, the trigonometric moment prob ..."
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Cited by 4 (4 self)
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In memory of Mark Grigoryevich Krein on the occasion of the 100th anniversary of his birth Abstract. The moment problem as formulated by Krein and Nudel’man is a beautiful generalization of several important classical moment problems, including the power moment problem, the trigonometric moment problem and the moment problem arising in NevanlinnaPick interpolation. Motivated by classical applications and examples, in both finite and infinite dimensions, we recently formulated a new version of this problem that we call the moment problem for positive rational measures. The formulation reflects the importance of rational functions in signals, systems and control. While this version of the problem is decidedly nonlinear, the basic tools still rely on convexity. In particular, we present a solution to this problem in terms of a nonlinear convex optimization problem that generalizes the maximum entropy approach used in several classical special cases.
Identification of rational spectral densities using orthonormal basis functions, in: The
 Proceedings of Symposium on System Identification
, 2003
"... Abstract: This paper gives an algorithm for identifying spectral densities using orthonormal basis functions. Mathematically, this amounts to identifying a timeinvariant linear SISO system with the additional constraint that the transfer function should be positivereal. Thus, we solve the longst ..."
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Cited by 3 (0 self)
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Abstract: This paper gives an algorithm for identifying spectral densities using orthonormal basis functions. Mathematically, this amounts to identifying a timeinvariant linear SISO system with the additional constraint that the transfer function should be positivereal. Thus, we solve the longstanding problem of how to incorporate this positivity constraint while using orthonormal basis functions. The procedure is a variant of the THREE algorithm introduced by Byrnes, Georgiou and Lindquist. The relation between and numerical properties of the proposed and the THREE algorithms are discussed. The orthonormal basis functions are better scaled for a concentrated pole selection in the basis, which increases the accuracy of the estimates. A numerical example which highlights this phenomenon and illustrates the algorithm is given.