Four Deviations Suffice for Rank 1 Matrices

@article{Kyng2019FourDS,
  title={Four Deviations Suffice for Rank 1 Matrices},
  author={Rasmus Kyng and Kyle Luh and Z. Song},
  journal={ArXiv},
  year={2019},
  volume={abs/1901.06731}
}
  • Rasmus Kyng, Kyle Luh, Z. Song
  • Published 2019
  • Mathematics, Computer Science
  • ArXiv
  • We prove a matrix discrepancy bound that strengthens the famous Kadison-Singer result of Marcus, Spielman, and Srivastava. Consider any independent scalar random variables $\xi_1, \ldots, \xi_n$ with finite support, e.g. $\{ \pm 1 \}$ or $\{ 0,1 \}$-valued random variables, or some combination thereof. Let $u_1, \dots, u_n \in \mathbb{C}^m$ and $$ \sigma^2 = \left\| \sum_{i=1}^n \text{Var}[ \xi_i ] (u_i u_i^{*})^2 \right\|. $$ Then there exists a choice of outcomes $\varepsilon_1,\ldots… CONTINUE READING
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