AI 中文总结
研究\(1 < p < \infty\)时\(L^p\)正则化及熵正则化最优传输的小正则化极限，通过新方法推导渐近性，推广现有结果并建立联系，明确了一阶和二阶渐近性。
AI 中文摘要
我们研究了\(1 < p < \infty\)时\(L^p\)正则化最优传输以及熵正则化最优传输（EOT）的小正则化极限。在对源测度和目标测度的温和假设下，明确确定了精确的一阶（分别为二阶）渐近性。我们的工作推广了二次和熵正则化的现有结果，并通过\(p\in(1,2)\)的自然插值将它们联系起来。我们通过一种新颖的方法以统一的方式推导所有这些渐近性，该方法将最优轮廓的局部计算与边际约束的全局执行分开：凸对偶性导致熵的高斯轮廓和\(L^p\)正则化的Barenblatt轮廓，而量化构造将这些局部轮廓转化为耦合。
英文摘要
We study the small-regularization limit for $L^p$-regularized optimal transport with $1<p<\infty$ and for entropically regularized optimal transport (EOT). The exact first-order (respectively, second-order) asymptotics are determined explicitly under mild assumptions on the source and target measures. Our work generalizes the existing results for quadratic and entropic regularization, and connects them by a natural interpolation via $p\in(1,2)$. We derive all these asymptotics in a unified manner by a novel approach that separates the local computation of the optimal profile from the global enforcement of the marginal constraints: convex duality leads to Gaussian profiles for entropy and Barenblatt profiles for $L^p$-regularization, while a quantization construction turns these local profiles into couplings.
Comments50 pages
