On a new method for controlling the entire spectrum in the problem of column subset selection
Résumé
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. Constructive approaches have been proposed lately, mainly sparked by the work of Batson, Spielman and Srivastava. The column selection problem we consider in this paper is concerned with extracting a well conditioned submatrix, and more precisely, a matrix with all its singular values being contained in the interval [1 − ε, 1 + ε]. Such results are known to have far reaching connections with many fields in mathematics and engineering. Our main contribution is a new deterministic method that achieves the same order R for the number of selected columns as in Bourgain and Tzafriri's original Theorem, up to a log(R) multiplicative factor. Our analysis is elementary and shows how a simple eigenvalue perturbation argument can lead to an intuitive and very short proof. We also obtain individual lower and upper bounds for each singular value of the extracted matrix.
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