Upper and Lower Bounds for Matrix Discrepancy

The aim of this paper is to study the matrix discrepancy problem. Assume that $$\xi _1,\ldots ,\xi _n$$ are independent scalar random variables with finite support and $${\textbf{u}}_1,\ldots ,{\textbf{u}}_n\in {\mathbb C}^d$$ . Let $${\mathcal C}_0$$ be minimal constant for which following holds: $$\begin{aligned} \textrm{Disc}({\textbf{u}}_1{\textbf{u}}_1^*,\ldots ,{\textbf{u}}_n{\textbf{u}}_n^*; \xi _n)\,\,:=\,\,\min _{\varepsilon _1\in {\mathcal S}_1,\ldots ,\varepsilon _n\in S}_n}\bigg \Vert \sum _{i=1}^n\mathbb {E}[\xi _i]{\textbf{u}}_i{\textbf{u}}_i^*-\sum _{i=1}^n\varepsilon _i{\textbf{u}}_i{\textbf{u}}_i^*\bigg \le C}_0\cdot \sigma , \end{aligned}$$ where $$\sigma ^2 = \big _{i=1}^n \text{ Var }[\xi _i]({\textbf{u}}_i{\textbf{u}}_i^*)^2\big $$ S}_j$$ denotes _j, j=1,\ldots ,n$$ Motivated by technology developed Bownik, Casazza, Marcus, Speegle [7], we prove C}_0\le 3$$ This improves Kyng, Luh Song’s method 4$$ [21]. For case $$\{{\textbf{u}}_i\}_{i=1}^n\subset a unit-norm tight frame n\le 2d-1$$ Rademacher variables, present exact value $$\textrm{Disc}({\textbf{u}}_1{\textbf{u}}_1^*,\ldots _n)=\sqrt{\frac{n}{d}}\cdot implies C}_0\ge \sqrt{2}$$