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An elementary approach to the problem of column selection in a rectangular matrix

Chretien, S; Dares, S (2018) An elementary approach to the problem of column selection in a rectangular matrix. In: Lecture Notes in Computer Science - Machine Learning, Optimization and Big Data. Springer, pp. 234-243. ISBN 9783319729251

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The problem of extracting a well conditioned submatrix from any rectangular matrix (with normalized columns) has been studied for some time in functional and harmonic analysis. In the seminal work of Bourgain and Tzafriri and many subsequent improvements, methods using random column selection were considered. More constructive approaches have been proposed recently, largely impulsed by the work of Bateson, Spielman and Srivastava. The column selection problem we consider in this paper is concerned with extracting a well conditioned submatrix, i.e. a matrix whose singular values all lie in [1-ε,1+ε]. We provide individual lower and upper bounds for each singular value of the extracted matrix at the price of conceding only one log factor in the number of columns, when compared to the Restricted Invertibility Theorem of Bourgain and Tzafriri. Our method is fully constructive and the proof is short and elementary.

Item Type: Book Chapter/Section
Subjects: Mathematics and Scientific Computing > Signal Processing
Divisions: Data Science
Publisher: Springer
Last Modified: 18 Apr 2018 14:44
URI: http://eprintspublications.npl.co.uk/id/eprint/7855

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