  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 $\sigma^2 = \big\|\sum_{i=1}^n
\mathbf{Var}[\xi_i](\mathbf{u}_i\mathbf{u}_i^*)^2\big\|$ 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
$\mathcal{C}_0\geq \sqrt{2}$.
