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Krylov subspace estimation. (English) Zbl 0980.65045

Authors’ abstract: Computing the linear least-squares estimate of a high-dimensional random quantity given noisy data requires solving a large system of linear equations. In many situations, one can solve this system efficiently using a Krylov subspace method, such as the conjugate gradient (CG) algorithm. Computing the estimation error variances is a more intricate task. It is difficult because the error variances are the diagonal elements of a matrix expression involving the inverse of a given matrix.
This paper presents a method for using the conjugate search directions generated by the CG algorithm to obtain a convergent approximation to the estimation error variances. The algorithm for computing the error variances falls out naturally from a new estimation-theoretic interpretation of the CG algorithm. This paper discusses this interpretation and convergence issues and presents numerical examples. The examples include a \(10^5\)-dimensional estimation problem from oceanography.
Reviewer’s remark: He wonders that the editors admitted this title since it does not give any information about the contents of the paper.

MSC:

65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F10 Iterative numerical methods for linear systems
62F10 Point estimation
62J05 Linear regression; mixed models
65C60 Computational problems in statistics (MSC2010)
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