AI 中文总结
该研究推导了高斯KL-UOT的高维随机矩阵极限，得到了脊乘积谱收敛、特征值极限等结果，为不平衡最优传输的高维分析提供了理论支撑。
AI 中文摘要
我们研究高斯Kullback-Leibler不平衡最优传输（KL-UOT）的高维随机矩阵极限。在边际惩罚相等的情况下，协方差作用具有精确的对数行列式表示，该表示基于非线性脊乘积，以及一个正半定扩展，该扩展在任意长宽比下均保持有限。对于独立实Wishart样本，强渐近自由性给出了极限自由乘法卷积和脊乘积谱的几乎必然Hausdorff收敛；独立Haar方向产生了变形总体的相应一阶极限。在对称非奇异单位Wishart模型中，我们推导了显式η变换和低次代数方程，该方程用于选择物理分支并确定支撑区间、平方根边缘以及极端特征值极限。我们进一步获得了有限四阶矩下的全长宽比单样本Marchenko–Pastur极限、c<1时的实高斯Bai–Silverstein波动，以及联合随机矩阵/惩罚极限，该极限表明样本协方差噪声产生临界尺度τ_p∝p。
英文摘要
We study high-dimensional random-matrix limits of Gaussian Kullback--Leibler unbalanced optimal transport (KL-UOT). Under equal marginal penalties, the covariance action admits an exact log-determinant representation in terms of a nonlinear ridge product, together with a positive-semidefinite extension that remains finite at arbitrary aspect ratios. For independent real Wishart samples, strong asymptotic freeness gives the limiting free multiplicative convolution and almost-sure Hausdorff convergence of the ridge-product spectrum; independent Haar orientations yield the corresponding first-order limit for deformed populations. In the symmetric nonsingular identity-Wishart model, we derive an explicit $η$-transform and a low-degree algebraic equation that select the physical branch and determine the support interval, square-root edges, and extreme-eigenvalue limits. We further obtain all-aspect one-sample Marchenko--Pastur limits under finite fourth moments, real-Gaussian Bai--Silverstein fluctuations for $c<1$, and a joint random-matrix/penalty limit showing that sample-covariance noise produces the critical scale $τ_p\asymp p$.
