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Analysis of the Cholesky decomposition of a semidefinite matrix
 in Reliable Numerical Computation
, 1990
"... Perturbation theory is developed for the Cholesky decomposition of an n × n symmetric positive semidefinite matrix A of rank r. The matrix W = A −1 11 A12 is found to play a key role in the perturbation bounds, where A11 and A12 are r × r and r × (n − r) submatrices of A respectively. A backward er ..."
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Cited by 65 (4 self)
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Perturbation theory is developed for the Cholesky decomposition of an n × n symmetric positive semidefinite matrix A of rank r. The matrix W = A −1 11 A12 is found to play a key role in the perturbation bounds, where A11 and A12 are r × r and r × (n − r) submatrices of A respectively. A backward
Positive SemiDefinite Matrix Multiplicative Error Models
, 2007
"... Positive semidefinite matrices arise in a number of interesting contexts, most notably in covariance modeling and forecasting but also in conditional duration and hazard models. Most existing covariance models are constructed as Multivariate GARCH models which use the distribution of returns to est ..."
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to estimate any unknown parameters in the conditional covariance. This paper introduced the positive semidefinite matrix multiplicative error model as a generalization of MVGARCH models to allow the modeling of positive semidefinite matrices which do not have a representation using returns. The paper
RANDOMIZED ALGORITHMS FOR ESTIMATING THE TRACE OF AN IMPLICIT SYMMETRIC POSITIVE SEMIDEFINITE MATRIX
"... Abstract. Finding the trace of an explicit matrix is a simple operation. But there are applications areas where one need to compute the trace of an implicit matrix. In this paper we explore the use of MonteCarlo simulation for estimating the trace of an implicit symmetric positive semide nite matr ..."
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Cited by 21 (0 self)
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Abstract. Finding the trace of an explicit matrix is a simple operation. But there are applications areas where one need to compute the trace of an implicit matrix. In this paper we explore the use of MonteCarlo simulation for estimating the trace of an implicit symmetric positive semide nite
Randomized Algorithms for Estimating the Trace of an Implicit Symmetric Positive SemiDefinite Matrix
"... We analyze the convergence of randomized trace estimators. Starting at 1989, several algorithms have been proposed for estimating the trace of a matrix by 1 ∑Mi=1 M zT i Azi, wheretheziare random vectors; different estimators use different distributions for the zis, all of which lead to E ( 1 ∑Mi=1 ..."
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We analyze the convergence of randomized trace estimators. Starting at 1989, several algorithms have been proposed for estimating the trace of a matrix by 1 ∑Mi=1 M zT i Azi, wheretheziare random vectors; different estimators use different distributions for the zis, all of which lead to E ( 1 ∑Mi=1
Learning the Kernel Matrix with SemiDefinite Programming
, 2002
"... Kernelbased learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information ..."
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Cited by 780 (22 self)
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problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semidefinite programming (SDP) techniques. When applied
Algorithms for Nonnegative Matrix Factorization
 In NIPS
, 2001
"... Nonnegative matrix factorization (NMF) has previously been shown to be a useful decomposition for multivariate data. Two different multiplicative algorithms for NMF are analyzed. They differ only slightly in the multiplicative factor used in the update rules. One algorithm can be shown to minim ..."
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Cited by 1230 (5 self)
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Nonnegative matrix factorization (NMF) has previously been shown to be a useful decomposition for multivariate data. Two different multiplicative algorithms for NMF are analyzed. They differ only slightly in the multiplicative factor used in the update rules. One algorithm can be shown
Stochastic Perturbation Theory
, 1988
"... . In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a firstorder perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating the variatio ..."
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Cited by 886 (35 self)
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. In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a firstorder perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating
Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization
 SIAM Journal on Optimization
, 1993
"... We study the semidefinite programming problem (SDP), i.e the problem of optimization of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First we review the classical cone duality as specialized to S ..."
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Cited by 557 (12 self)
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We study the semidefinite programming problem (SDP), i.e the problem of optimization of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First we review the classical cone duality as specialized
Global Optimization with Polynomials and the Problem of Moments
 SIAM Journal on Optimization
, 2001
"... We consider the problem of finding the unconstrained global minimum of a realvalued polynomial p(x) : R R, as well as the global minimum of p(x), in a compact set K defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear mat ..."
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Cited by 569 (47 self)
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matrix inequality (LMI) problems. A notion of KarushKuhnTucker polynomials is introduced in a global optimality condition. Some illustrative examples are provided. Key words. global optimization, theory of moments and positive polynomials, semidefinite programming AMS subject classifications. 90C22
Results 1  10
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243,315