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Robust Solutions To LeastSquares Problems With Uncertain Data
, 1997
"... . We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpret ..."
Abstract

Cited by 144 (12 self)
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. We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution, and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomialtime using semidefinite programming (SDP). We also consider the case when A; b are rational functions of an unknownbutbounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worstcase residual. We provide numerical examples, including one from robust identification and one from robust interpolation. Key Words. Leastsquares, uncertainty, robustness, secondorder cone...
Inversion Error, Condition Number, And Approximate Inverses Of Uncertain Matrices
 Inverses of Uncertain Matrices. Linear Algebra and its Applications
, 2000
"... The classical condition number is a very rough measure of the effect of perturbations on the inverse of a square matrix. First, it assumes the perturbation is infinitesimally small. Second, it does not take into account the perturbation structure (e.g., Vandermonde). Similarly, the classical notion ..."
Abstract

Cited by 6 (1 self)
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The classical condition number is a very rough measure of the effect of perturbations on the inverse of a square matrix. First, it assumes the perturbation is infinitesimally small. Second, it does not take into account the perturbation structure (e.g., Vandermonde). Similarly, the classical notion of inverse of a matrix neglects the possibility of large, structured perturbations. We define a new quantity, the structured maximal inversion error, that takes into account both structure and non necessarily small perturbation size. When the perturbation is infinitesimal, we obtain a "structured condition number". We introduce the notion of approximate inverse, as a matrix that best approximates the inverse of a matrix with structured perturbations, when the perturbation varies in a given range. For a wide class of perturbation structures, we show how to use (convex) semidefinite programming to compute bounds on on the structured maximal inversion error and structured condition number, and compute an approximate inverse. The results are exact when the perturbation is "unstructured"we then obtain an analytic expression for the approximate inverse. When the perturbation is unstructured and additive, we recover the classical condition number; the approximate inverse is the operator related to the Total Least Squares (orthogonal regression) problem.
Robust Solutions To LeastSquares Problems With Uncertain Data
, 1997
"... We consider leastsquares problems where the coefficient matrices A, b are unknown but bounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interprete ..."
Abstract
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We consider leastsquares problems where the coefficient matrices A, b are unknown but bounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomialtime using semidefinite programming (SDP). We also consider the case when A, b are rational functions of an unknownbutbounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worstcase residual. We provide numerical examples, including one from robust identification and one from robust interpolation.