Minimizing the condition number of a positive definite matrix by completion
نویسنده
چکیده
We consider the problem of minimizing the spectral condition number of a positive definite matrix by completion: min{cond( [ A BH B X ] ) : [ A BH B X ] positive definite}, where A is an n × n Hermitian positive definite matrix, B a p × n matrix and X is a free p× p Hermitian matrix. We reduce this problem to an optimization problem for a convex function in one variable. Using the minimal solution of this problem we characterize the complete set of matrices that give the minimum condition number. Mathematics Subject Classification: 65F35, 15A12
منابع مشابه
Minimizing the Condition Number of a Positive Deenite Matrix by Completion
We consider the problem of minimizing the spectral condition number of a positive deenite matrix by completion: minfcond(positive deeniteg; where A is an n n Hermitian positive deenite matrix, B a p n matrix and X is a free p p Hermitian matrix. We reduce this problem to an optimization problem for a convex function in one variable. Using the minimal solution of this problem we characterize the...
متن کاملMinimizing Condition Number via Convex Programming
In this paper we consider minimizing the spectral condition number of a positive semidefinite matrix over a nonempty closed convex set Ω. We show that it can be solved as a convex programming problem, and moreover, the optimal value of the latter problem is achievable. As a consequence, when Ω is positive semidefinite representable, it can be cast into a semidefinite programming problem. We the...
متن کاملInteger Factorization of a Positive-Definite Matrix
This paper establishes that every positive-definite matrix can be written as a positive linear combination of outer products of integer-valued vectors whose entries are bounded by the geometric mean of the condition number and the dimension of the matrix.
متن کاملPositive Semidefinite Matrix Completions on Chordal Graphs and Constraint Nondegeneracy in Semidefinite Programming
Let G = (V, E) be a graph. In matrix completion theory, it is known that the following two conditions are equivalent: (i) G is a chordal graph; (ii) Every G-partial positive semidefinite matrix has a positive semidefinite matrix completion. In this paper, we relate these two conditions to constraint nondegeneracy condition in semidefinite programming and prove that they are each equivalent to (...
متن کاملLarge-scale log-determinant computation through stochastic Chebyshev expansions
Logarithms of determinants of large positive definite matrices appear ubiquitously in machine learning applications including Gaussian graphical and Gaussian process models, partition functions of discrete graphical models, minimum-volume ellipsoids, metric learning and kernel learning. Log-determinant computation involves the Cholesky decomposition at the cost cubic in the number of variables,...
متن کامل