发表机构
UNC Chapel Hill（北卡罗来纳大学教堂山分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在arXiv:2607.24164v1相关定理的前提下，证明了Jiang-Lee-Vempala关于对数凹随机向量内积的Wasserstein距离的猜想及矩形双线性型推广，构造斯坦核得到相关距离界并发展了对应证明技术。
AI 中文摘要
Jiang、Lee和Vempala猜想：若$X,Y\in\mathbb{R}^n$是独立的各向同性对数凹随机向量，则$W_2(L(\langle X,Y\rangle),N(0,n))$有界于一个普适常数。在arXiv:2607.24164v1的定理1.2和2.5的前提下，我们证明了该猜想及矩形双线性型的推广。对独立的各向同性对数凹$X\in\mathbb{R}^m$、$Y\in\mathbb{R}^n$和非零$B\in\mathbb{R}^{m\times n}$，定义$r_4(B)=\frac{(\operatorname{Tr}(B^\top B))^2}{\operatorname{Tr}((B^\top B)^2)}$。我们为$X^\top B Y/\\|B\\|_F$构造非负标量斯坦核，其平方$L^2$偏差至多为$20/r_4(B)$，因此得到其与$N(0,1)$的平方2- Wasserstein距离的相同界。证明过程发展了矩映射斯坦核的迹可观测量的精确协方差恒等式与亏空分解，以及正斯坦核在对数凹近似下的稳定性定理。取$B=I_n$可得$W_2^2\left(L\left(\frac{\langle X,Y\rangle}{\sqrt{n}}\right),N(0,1)\right)\leq\frac{20}{n}$，这就是Jiang-Lee-Vempala猜想。
英文摘要
Jiang, Lee, and Vempala conjectured that if $X,Y\in\mathbb{R}^n$ are independent isotropic log-concave random vectors, then $W_2(L(\langle X,Y\rangle),N(0,n))$ is bounded by a universal constant. Subject to Theorems 1.2 and 2.5 of arXiv:2607.24164v1, we prove this conjecture and a rectangular bilinear-form extension. For independent isotropic log-concave $X\in\mathbb{R}^m$, $Y\in\mathbb{R}^n$, and nonzero $B\in\mathbb{R}^{m\times n}$, put \[ r_4(B)=\frac{(\operatorname{Tr}(B^\top B))^2} {\operatorname{Tr}((B^\top B)^2)}. \] We construct a nonnegative scalar Stein kernel for $X^\top B Y/\|B\|_F$ whose squared $L^2$ discrepancy is at most $20/r_4(B)$, and consequently obtain the same bound for squared $2$-Wasserstein distance to $N(0,1)$. The proof develops an exact covariance identity and deficit decomposition for trace observables of moment-map Stein kernels, together with a stability theorem for positive Stein kernels under log-concave approximation. Taking $B=I_n$ yields \[ W_2^2\left(L\left(\frac{\langle X,Y\rangle}{\sqrt{n}}\right),N(0,1)\right)\leq\frac{20}{n}, \] which is the Jiang--Lee--Vempala conjecture.
