The aim of this paper is to study the matrix discrepancy problem. Assume that ξ1,…,ξn\documentclass[12pt]{minimal}
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\begin{document}$$\xi _1,\ldots ,\xi _n$$\end{document} are independent scalar random variables with finite support and u1,…,un∈Cd\documentclass[12pt]{minimal}
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\begin{document}$${\textbf{u}}_1,\ldots ,{\textbf{u}}_n\in {\mathbb C}^d$$\end{document}. Let C0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal C}_0$$\end{document} be the minimal constant for which the following holds: Disc(u1u1∗,…,unun∗;ξ1,…,ξn):=minε1∈S1,…,εn∈Sn‖∑i=1nE[ξi]uiui∗-∑i=1nεiuiui∗‖≤C0·σ,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \textrm{Disc}({\textbf{u}}_1{\textbf{u}}_1^*,\ldots ,{\textbf{u}}_n{\textbf{u}}_n^*; \xi _1,\ldots ,\xi _n)\,\,:=\,\,\min _{\varepsilon _1\in {\mathcal S}_1,\ldots ,\varepsilon _n\in {\mathcal S}_n}\bigg \Vert \sum _{i=1}^n\mathbb {E}[\xi _i]{\textbf{u}}_i{\textbf{u}}_i^*-\sum _{i=1}^n\varepsilon _i{\textbf{u}}_i{\textbf{u}}_i^*\bigg \Vert \le {\mathcal C}_0\cdot \sigma , \end{aligned}$$\end{document}where σ2=‖∑i=1nVar[ξi](uiui∗)2‖\documentclass[12pt]{minimal}
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\begin{document}$$\sigma ^2 = \big \Vert \sum _{i=1}^n \text{ Var }[\xi _i]({\textbf{u}}_i{\textbf{u}}_i^*)^2\big \Vert $$\end{document} and Sj\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal S}_j$$\end{document} denotes the support of ξj,j=1,…,n\documentclass[12pt]{minimal}
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\begin{document}$$\xi _j, j=1,\ldots ,n$$\end{document}. Motivated by the technology developed by Bownik, Casazza, Marcus, and Speegle [7], we prove C0≤3\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal C}_0\le 3$$\end{document}. This improves Kyng, Luh and Song’s method with which C0≤4\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal C}_0\le 4$$\end{document} [21]. For the case where {ui}i=1n⊂Cd\documentclass[12pt]{minimal}
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\begin{document}$$\{{\textbf{u}}_i\}_{i=1}^n\subset {\mathbb C}^d$$\end{document} is a unit-norm tight frame with n≤2d-1\documentclass[12pt]{minimal}
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\begin{document}$$ n\le 2d-1$$\end{document} and ξ1,…,ξn\documentclass[12pt]{minimal}
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\begin{document}$$\xi _1,\ldots ,\xi _n$$\end{document} are independent Rademacher random variables, we present the exact value of Disc(u1u1∗,…,unun∗;ξ1,…,ξn)=nd·σ\documentclass[12pt]{minimal}
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\begin{document}$$\textrm{Disc}({\textbf{u}}_1{\textbf{u}}_1^*,\ldots ,{\textbf{u}}_n{\textbf{u}}_n^*; \xi _1,\ldots ,\xi _n)=\sqrt{\frac{n}{d}}\cdot \sigma $$\end{document}, which implies C0≥2\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal C}_0\ge \sqrt{2}$$\end{document}.
