Upper and Lower bounds for matrix discrepancy (2006.12083v2)

Abstract: 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 $\mathbf{u}1,\ldots,\mathbf{u}_n\in \mathbb{C}^d$. Let $\mathcal{C}_0$ be the minimal constant for which the following holds: [ {\rm Disc}(\mathbf{u}_1\mathbf{u}_1^,\ldots,\mathbf{u}_n\mathbf{u}_n^; \xi_1,\ldots,\xi_n)\,\,:=\,\,\min{\varepsilon_1\in \mathcal{S}1,\ldots,\varepsilon_n\in \mathcal{S}_n}\bigg|\sum{i=1}^n\mathbb{E}[\xi_i]\mathbf{u}i\mathbf{u}_i^*-\sum{i=1}^n\varepsilon_i\mathbf{u}_i\mathbf{u}_i^*\bigg|\leq \mathcal{C}0\cdot\sigma, ] where $\sigma^2 = \big|\sum{i=1}^n \mathbf{Var}\xi_i^2\big|$ and $\mathcal{S}j$ denotes the support of $\xi_j, j=1,\ldots,n$. Motivated by the technology developed by Bownik, Casazza, Marcus, and Speegle, we prove $\mathcal{C}_0\leq 3$. This improves Kyng, Luh and Song's method with which $\mathcal{C}_0\leq 4$. For the case where ${\mathbf{u}_i}{i=1}^n\subset \mathbb{C}^d$ is a unit-norm tight frame with $ n\leq 2d-1$ and $\xi_1,\ldots,\xi_n$ are independent Rademacher random variables, we present the exact value of ${\rm Disc}(\mathbf{u}_1\mathbf{u}_1^,\ldots,\mathbf{u}_n\mathbf{u}_n^; \xi_1,\ldots,\xi_n)=\sqrt{\frac{n}{d}}\cdot\sigma$, which implies $\mathcal{C}_0\geq \sqrt{2}$.