发表机构
University of California, San Diego（加州大学圣地亚哥分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了两因子Recht-Re下界对每个n都是尖锐的，构造了达到界的秩一矩阵族，分类了等式情形，并展示了强凸二次目标中极值负特征值的实现。
AI 中文摘要
当随机优化方法的连续步骤使用不同数据点时，会出现正半定矩阵的乘积。尽管每个单独的矩阵都是正半定的，它们的平均值可能具有负特征值。Lai和Lim在ICML 2020上证明了当$A_i\succeq0$且$\sum_i A_i\preceq nI$时，$\sum_{i\ne j}A_iA_j\succeq-\frac{1}{4}n(n-1)I$，并猜想该常数对每个$n\ge2$都是尖锐的。我们给出了一个显式的$n$个秩一矩阵族，在维度$n$中对每个$n$都达到该界，从而证明了尖锐性陈述。该构造将标准正交基的一个方向压缩了因子$1/\sqrt{2}$。我们还给出了已知不等式的简短、自包含的证明，并对所有等式情形进行了分类：每个极值元都包含一个与构造等价的不变$n$维块。因此，维度$n$是实现等式所必需的，且正定矩阵接近但从未达到下端点。最后，强凸二次目标将极值负特征值实现为两步期望梯度下降更新。
英文摘要
Products of positive semidefinite matrices arise when successive steps of a randomized optimization method use different data points. Their average can have a negative eigenvalue even though every individual matrix is positive semidefinite. Lai and Lim proved at ICML 2020 that $\sum_{i\ne j}A_iA_j\succeq-\frac{1}{4}n(n-1)I$ whenever $A_i\succeq0$ and $\sum_i A_i\preceq nI$, and conjectured that the constant is sharp for every $n\ge2$. We give an explicit family of $n$ rank-one matrices in dimension $n$ that attains this bound for every $n$, proving that sharpness statement. The construction compresses one direction of an orthonormal basis by a factor $1/\sqrt{2}$. We also give a short, self-contained proof of the known inequality and classify all equality cases: every extremizer contains a common invariant $n$-dimensional block equivalent to the construction. Consequently, dimension $n$ is necessary for equality, and positive definite matrices approach but never attain the lower endpoint. Finally, strongly convex quadratic objectives realize the extremal negative eigenvalue as a two-step expected gradient-descent update.
