TL;DR
This paper presents deterministic methods for selecting large submatrices with controlled singular values, with applications to ellipsoid contact points, paving problems, and matrix norm reduction, advancing understanding in matrix subset selection.
Contribution
It introduces a deterministic algorithm for extracting large submatrices with specific spectral properties, extending previous probabilistic results and addressing open questions.
Findings
Deterministic bounds on singular values of submatrices
Algorithm for partitioning matrices into almost isometric blocks
Partial solution to Naor's question on submatrix norm minimization
Abstract
Given a matrix U, using a deterministic method, we extract a "large" submatrix of U'(whose columns are obtained by normalizing those of U) and estimate its smallest and largest singular value. We apply this result to the study of contact points of the unit ball with its maximal volume ellipsoid. We consider also the paving problem and give a deterministic algorithm to partition a matrix into almost isometric blocks recovering previous results of Bourgain-Tzafriri and Tropp. Finally, we partially answer a question raised by Naor about finding an algorithm in the spirit of Batson-Spielman-Srivastava's work to extract a "large" square submatrix of "small" norm.
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