论文标题
矩阵差异的上限和下限
Upper and Lower bounds for matrix discrepancy
论文作者
论文摘要
本文的目的是研究矩阵差异问题。假设$ξ_1，\ ldots，ξ_n$是具有有限支持的独立标量随机变量，$ \ Mathbf {u} _1，\ ldots，\ Mathbf {u} _n \ in \ Mathbb {C}^d $。令$ \ Mathcal {C} _0 $为以下最小常数：\ [ {\ rm disc}（\ mathbf {u} _1 \ mathbf {u} _1^*，\ ldots，\ mathbf {u} _n \ mathbf {u} _nn^*; ξ_1，\ ldots，ξ_n）\，\，：= \，\，\ min _ {\ varepsilon_1 \ in \ Mathcal {s} _1，\ ldots，\ varepsilon_n_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} \ Mathcal {C} _0 \cdotσ，\]其中$σ^2 = \ big \ | \ sum_ {i = 1}^n \ Mathbf {var} [var} [ξ_i]（\ Mathbf { $ \ MATHCAL {S} _J $表示$ξ_j的支持，j = 1，\ ldots，n $。由Bownik，Casazza，Marcus和Speegle开发的技术激励，我们证明了$ \ Mathcal {C} _0 \ leq 3 $。这改善了Kyng，Luh和Song的方法，其中$ \ 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 ${\rm disc}（\ MathBf {U} _1 \ MathBf {u} _1^*，\ ldots，\ MathBf {U} _n \ Mathbf {U} _n^*; emed_1;ξ_1，\ ldots，\ ldots，\ ldots，\ ldots，之一含义$ \ Mathcal {C} _0 \ geq \ sqrt {2} $。
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 $\mathcal{C}_0\geq \sqrt{2}$.
