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-rhologsig...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 52; no. 3; pp. 1052 - 1066
Main Author: Georgiou, T.T.
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.03.2006
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algebra
Applied sciences
Covariance realization
Entropy
Exact sciences and technology
Functionals
Information theory
Information, signal and communications theory
Inverse problems
Mathematical functions
moment problem
multidimensional
Multidimensional systems
Multivariable
Operators
Polarization
Probability distribution
quantum entropy
Radar scattering
Sensor arrays
Sensor phenomena and characterization
Signal and communications theory
Signal representation. Spectral analysis
Signal, noise
Spectral analysis
Telecommunications and information theory
Second order
Quantum system
quantum entropy
Riemann metric
multivariable
Order statistic
spectral analysis
Entropy
Inverse problem
Statistical method
Covariance
Spectrum analysis
Homotopy
Moment problem
Renormalization
Signal analysis
multidimensional
Covariance realization
Power spectrum
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary: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
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2005.864422