复高斯矩阵平均奇异值的非渐近界
Non-asymptotic bounds for the average singular value of a complex Gaussian matrix
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中文总结 AI 辅助
本文针对复高斯矩阵平均奇异值，通过连续对偶Hahn表示和奇异性分析，给出其相邻差及自身在所有维度的非渐近上下界，并匹配渐近展开首项。
中文摘要 AI 辅助
设$G_{d}$为具有独立标准复高斯元素的$d \times d$矩阵，令$\alpha_{\mathbb{C}}(d)$为$G_{d}/\sqrt{d}$的期望平均奇异值，并设$\Delta_d:= \alpha_{\mathbb{C}}(d)-\alpha_{\mathbb{C}}(d+1)$。统计量$\alpha_{\mathbb{C}}(d)$具有变分表示，即酉群上的期望归一化最大值，并控制酉群上小Grothendieck问题及相关酉配准问题的近似保证。我们获得$\Delta_d$的严格正下界和上界，两者在所有维度均成立，同时给出$\alpha_{\mathbb{C}}(d)$在Marchenko–Pastur极限附近的相应界。这些界与两个量的完整渐近展开的尖锐首项行为相匹配，其系数可显式计算。证明结合了由$Y_d = d^{3/2}\alpha_{\mathbb{C}}(d)$的连续对偶Hahn表示得到的三项递推关系，以及对底层Laguerre矩生成函数的奇异性分析。
英文摘要
Let $G_{d}$ be a $d \times d$ matrix with independent standard complex Gaussian entries, let $α_{\mathbb{C}}(d)$ be the expected average singular value of $G_{d}/\sqrt{d}$, and set $Δ_d := α_{\mathbb{C}}(d)-α_{\mathbb{C}}(d+1)$. The statistic $α_{\mathbb{C}}(d)$ admits a variational representation as an expected normalized maximum over the unitary group and governs approximation guarantees for the little Grothendieck problem over the unitary group and related unitary registration problems. We obtain a strictly positive lower bound and an upper bound for $Δ_d$, both valid in every dimension, together with corresponding bounds for $α_{\mathbb{C}}(d)$ around the Marchenko--Pastur limit. These bounds match the sharp leading behavior of complete asymptotic expansions for both quantities, whose coefficients are explicitly computable. The proof combines a three-term recurrence for $Y_d = d^{3/2}α_{\mathbb{C}}(d)$, obtained from its continuous dual Hahn representation, along with singularity analysis of the underlying Laguerre moment generating function.
发表机构
- University of Vienna（维也纳大学）
- University of Texas at Austin（德克萨斯大学奥斯汀分校）
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