巴兰姜矩阵问题的下限

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis Pub Date : 2024-08-06 DOI:10.1016/j.acha.2024.101696

Afonso S. Bandeira , Dustin G. Mixon , Stefan Steinerberger

引用次数: 0

摘要

我们证明了一个正半有限矩阵 A∈Rn×n 的存在性，即任何分解为秩-1 矩阵的矩阵都必须具有较大 ℓ1-norm 的因子，更确切地说∑kxkxk⁎=A⇒∑k‖xk‖12≥cn‖A1，其中 c 与 n 无关。这就为巴兰姜矩阵问题提供了一个下界。这种构造是概率性的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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A lower bound for the Balan–Jiang matrix problem

We prove the existence of a positive semidefinite matrix $A \in R^{n \times n}$ such that any decomposition into rank-1 matrices has to have factors with a large $ℓ^{1} -$ norm, more precisely $\sum_{k} x_{k} x_{k}^{⁎} = A \Rightarrow \sum_{k} {‖ x_{k} ‖}_{1}^{2} \geq c \sqrt{n} {‖ A ‖}_{1},$ where c is independent of n. This provides a lower bound for the Balan–Jiang matrix problem. The construction is probabilistic.

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来源期刊

Applied and Computational Harmonic Analysis 物理-物理：数学物理

CiteScore

5.40

自引率

4.00%

发文量

审稿时长

22.9 weeks

期刊介绍： Applied and Computational Harmonic Analysis (ACHA) is an interdisciplinary journal that publishes high-quality papers in all areas of mathematical sciences related to the applied and computational aspects of harmonic analysis, with special emphasis on innovative theoretical development, methods, and algorithms, for information processing, manipulation, understanding, and so forth. The objectives of the journal are to chronicle the important publications in the rapidly growing field of data representation and analysis, to stimulate research in relevant interdisciplinary areas, and to provide a common link among mathematical, physical, and life scientists, as well as engineers.