发表机构
Auburn University（奥本大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明当 $n\ge 2\lceil m/2 \rceil^2$ 时非交换AM-GM不等式成立，通过基于切比雪夫节点的顶点测度提取指标，并借助求积估计解决测度非正性的困难。
AI 中文摘要
在本文中，我们证明了当 $n\ge 2\lceil m/2 \rceil^2$ 时，非交换AM-GM不等式成立。该研究的动机来自[De Sa, Random reshuffling is not always better, NeurIPS2020]中构造的反例。证明基于布尔立方体上的切比雪夫节点构造了一个顶点测度，以提取不同的指标。主要困难在于该测度在非整数节点上不是正的。关键技术来自[Grigoriev, Complexity of Positivstellensatz proofs for the knapsack, Computational Complexity (2001)]，并最终将问题转化为求积估计。
英文摘要
In this note, we prove that the noncommutative AM-GM inequality holds if $n\ge 2\lceil m/2 \rceil^2$. The motivation comes from counterexamples constructed in [De Sa, Random reshuffling is not always better, NeurIPS2020]. The proof constructs a vertex measure based on the Chebyshev nodes on the Boolean cube to extract the distinct indices. The main difficulty is that the measure is not positive on non-integer nodes. The key technique comes from [Grigoriev, Complexity of Positivstellensatz proofs for the knapsack, Computational Complexity (2001)] and eventually transforms the problem into a quadrature estimate.
