The random paving property for uniformly bounded matrices
Abstract
This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison--Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of Bourgain and Tzafriri involves a long, delicate calculation. The new proof relies on the systematic use of symmetrization and (noncommutative) Khintchine inequalities to estimate the norms of some random matrices.
Cite
@article{arxiv.math/0612070,
title = {The random paving property for uniformly bounded matrices},
author = {Joel A. Tropp},
journal= {arXiv preprint arXiv:math/0612070},
year = {2014}
}
Comments
12 pages; v2 with cosmetic changes; v3 with corrections to Prop. 4; v4 with minor changes to text; v5 with correction to discussion of noncommutative Khintchine inequality; v6 with slight improvement to main theorem