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Matrix convex functions with applications to weighted centers 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 − log X 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 − log X 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 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.
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.
unknown title
, 2006
"... The maximum entropy ansatz in the absence of a time arrow: fractionalpole models ..."
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The maximum entropy ansatz in the absence of a time arrow: fractionalpole models
OF THE UNIVERSITY OF MINNESOTA BY
, 2011
"... It has been such an honor to work with him, a great mind whose knowledge and intellectual sharpness marvels me all the time. I appreciate the excellent example he provided as one who really enjoys learning and conducts research with a sincere heart. I am grateful not only for his guidance on my rese ..."
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It has been such an honor to work with him, a great mind whose knowledge and intellectual sharpness marvels me all the time. I appreciate the excellent example he provided as one who really enjoys learning and conducts research with a sincere heart. I am grateful not only for his guidance on my research and study, but also his encouragement and confidence in me, which is especially invaluable during my difficult times. This thesis would not have been possible without his guidance and help. This is also an opportunity for me to acknowledge the financial support from the National Science Foundation and the US Air Force Office of Scientific Research. I want to express my deep gratitude to my oral exam committee members: Professor Murti V. Salapaka for serving as the chair, Professor ZhiQuan (Tom) Luo for his insight and suggestions on the problem of spectral tracking, Professor Mihailo Jovanovic who kindly let me share the office with his students for all these years, and Professor Stergios I. Roumeliotis to whom I am especially grateful for his help and kindness towards my entire family. It has been my privilege to get to know Professor Robert R. Snapp at the University of Vermont. Without his encouragement and help during my twoyear study there, it would not