Upper and Lower bounds for matrix discrepancy

TL;DR

This paper establishes new bounds for the matrix discrepancy problem, improving previous bounds and providing exact values in specific cases, advancing understanding of matrix balancing with random variables.

Contribution

The paper proves an improved universal bound of 3 for matrix discrepancy and determines exact discrepancy values for certain tight frames with Rademacher variables.

Findings

Proved  as an upper bound for matrix discrepancy.

Calculated exact discrepancy for specific tight frames with Rademacher variables.

Established a lower bound of 2 for the discrepancy in certain cases.

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