复高斯随机矩阵平均奇异值的单调性
On the monotonicity of average singular values of complex Gaussian random matrices
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中文总结 AI 辅助
针对复高斯随机矩阵平均奇异值随维度减小的性质，补充证明并将其单调性从非负整数形状参数α推广至实α≥0，完善了相关猜想的结论。
中文摘要 AI 辅助
我们给出复高斯随机矩阵平均奇异值随维度减小这一事实的另一种证明。该性质由Bandeira、Kennedy和Singer（《Math. Program.》，2016）提出猜想，近期Hutník针对非负整数形状参数α证实，我们还将该单调性推广至实α≥0的情形。
英文摘要
We prove that the normalized $k$-th moment of the Laguerre unitary ensemble, as a function of the dimension, is decreasing for fixed $k\in(0,1)\cup (2,\infty)$ and increasing for fixed $k\in(1,2).$ As a corollary, this implies that the average singular value of complex Gaussian random matrix decreases with dimension. Conjectured by Bandeira, Kennedy, and Singer (Math. Program., 2016), this property was recently established by Hutnik, for non-negative integer shape parameter $α$. Our result extends this monotonicity to real $α\geq 0$.
发表机构
- University of Toronto（多伦多大学）
- Indian Institute of Science Education and Research Kolkata（印度科学教育研究所加尔各答分校）
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