Improved bounds in Weaver's ${\rm KS}_r$ conjecture for high rank positive semidefinite matrices
Abstract
Recently Marcus, Spielman and Srivastava proved Weaver's conjecture, which gives a positive solution to the Kadison-Singer problem. Cohen and Br\"and\'en independently extended this result to obtain the arbitrary-rank version of Weaver's conjecture. In this paper, we present a new bound in Weaver's conjecture for the arbitrary-rank case. To do that, we introduce the definition of -characteristic polynomials and employ it to improve the previous estimate on the largest root of the mixed characteristic polynomials. For the rank-one case, our bound agrees with the Bownik-Casazza-Marcus-Speegle's bound when and with the Ravichandran-Leake's bound when . For the higher-rank case, we sharpen the previous bounds from Cohen and from Br\"and\'en .
Keywords
Cite
@article{arxiv.2106.09205,
title = {Improved bounds in Weaver's ${\rm KS}_r$ conjecture for high rank positive semidefinite matrices},
author = {Zhiqiang Xu and Zili Xu and Ziheng Zhu},
journal= {arXiv preprint arXiv:2106.09205},
year = {2021}
}