arXiv 2006.120832020-06math.CO

矩阵差异的上界和下界

Upper and Lower bounds for matrix discrepancy

Jiaxin Xie, Zhiqiang Xu, Ziheng Zhu

摘要：

本文的目的是研究矩阵不等式问题。假设$ξ_1,\ldots,ξ_n$和$\mathbf{u}_1,\ldots,\mathbf{u}_n\in \mathbb{C}^d$是具有有限支撑的独立标量随机变量。设$\mathcal{C}_0$为最小常数，满足以下条件：\[ {\rm Disc}(\mathbf{u}_1\mathbf{u}_1^*,\ldots,\mathbf{u}_n\mathbf{u}_n^*; ξ_1,\ldots,ξ_n)\,\,:=\,\,\min_{\varepsilon_1\in \mathcal{S}_1,\ldots,\varepsilon_n\in \mathcal{S}_n}\bigg\|\sum_{i=1}^n\mathbb{E}[ξ_i]\mathbf{u}_i\mathbf{u}_i^*-\sum_{i=1}^n\varepsilon_i\mathbf{u}_i\mathbf{u}_i^*\bigg\|\leq \mathcal{C}_0\cdotσ, \]其中$σ^2 = \big\|\sum_{i=1}^n \mathbf{Var}[ξ_i](\mathbf{u}_i\mathbf{u}_i^*)^2\big\|$和$\mathcal{S}_j$表示$ξ_j, j=1,\ldots,n$的支持。由Bownik，Casazza，Marcus和Speegle开发的技术的激励下，我们证明了$\mathcal{C}_0\leq 3$。这改进了Kyng，Luh和Song的方法，其中$\mathcal{C}_0\leq 4$。对于$\{\mathbf{u}_i\}_{i=1}^n\subset \mathbb{C}^d$是单位范数紧框架，$ n\leq 2d-1$和$ξ_1,\ldots,ξ_n$是独立的Rademacher随机变量的情形，我们给出了${\rm Disc}(\mathbf{u}_1\mathbf{u}_1^*,\ldots,\mathbf{u}_n\mathbf{u}_n^*; ξ_1,\ldots,ξ_n)=\sqrt{\frac{n}{d}}\cdotσ$的精确值，这意味着$\mathcal{C}_0\geq \sqrt{2}$ .

英文摘要：

The aim of this paper is to study the matrix discrepancy problem. Assume that $ξ_1,\ldots,ξ_n$ 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: \[ {\rm Disc}(\mathbf{u}_1\mathbf{u}_1^*,\ldots,\mathbf{u}_n\mathbf{u}_n^*; ξ_1,\ldots,ξ_n)\,\,:=\,\,\min_{\varepsilon_1\in \mathcal{S}_1,\ldots,\varepsilon_n\in \mathcal{S}_n}\bigg\|\sum_{i=1}^n\mathbb{E}[ξ_i]\mathbf{u}_i\mathbf{u}_i^*-\sum_{i=1}^n\varepsilon_i\mathbf{u}_i\mathbf{u}_i^*\bigg\|\leq \mathcal{C}_0\cdotσ, \] where $σ^2 = \big\|\sum_{i=1}^n \mathbf{Var}[ξ_i](\mathbf{u}_i\mathbf{u}_i^*)^2\big\|$ and $\mathcal{S}_j$ denotes the support of $ξ_j, j=1,\ldots,n$. 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 $ξ_1,\ldots,ξ_n$ are independent Rademacher random variables, we present the exact value of ${\rm Disc}(\mathbf{u}_1\mathbf{u}_1^*,\ldots,\mathbf{u}_n\mathbf{u}_n^*; ξ_1,\ldots,ξ_n)=\sqrt{\frac{n}{d}}\cdotσ$, which implies $\mathcal{C}_0\geq \sqrt{2}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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