AI 中文总结
本文研究Tsallis正则化最优传输到KL极限的收敛性，通过变分重构和Γ-收敛证明，给出O(q-1)误差估计，阐明熵正则化与信息投影的关系。
AI 中文摘要
我们研究了在非负有界连续代价下，熵正则化最优传输的Tsallis到Kullback-Leibler（KL）极限。固定正则化参数ε>0，我们首先推导出Tsallis正则化最优传输在耦合集上的Tsallis信息投影的精确变分重构。该公式分离出一个显式修正项，从而解释了为何与KL情形不同，正则化传输问题与相应的信息投影问题并不完全一致。我们还建立了Tsallis信息投影的存在性和唯一性。然后，我们证明了在窄拓扑下，当q↓1时，Tsallis正则化泛函Γ-收敛到KL正则化泛函，同时其唯一极小元窄收敛。最后，我们获得了正则化最优传输值和相应信息投影值的O(q-1)阶显式误差估计。这些结果量化了从Tsallis正则化到经典KL设置的过渡，并阐明了对于1<q≤2，熵正则化与信息投影之间的关系。
英文摘要
We study the Tsallis-to-Kullback--Leibler (KL) limit for entropy-regularized optimal transport with nonnegative bounded continuous costs. Fixing the regularization parameter $\varepsilon > 0$, we first derive an exact variational reformulation of Tsallis-regularized optimal transport in terms of the Tsallis information projection onto the set of couplings. The formula isolates an explicit correction term and thereby explains why, unlike in the KL case, the regularized transport problem and the corresponding information projection problem do not coincide exactly. We also establish existence and uniqueness for the Tsallis information projection. We then prove, with respect to the narrow topology, the $Γ$-convergence of the Tsallis-regularized functionals to the KL-regularized functional as $q\downarrow1$, together with narrow convergence of their unique minimizers. Finally, we obtain explicit error estimates of order $O(q-1)$ for both the regularized optimal transport values and the associated information projection values. These results quantify the passage from Tsallis regularization to the classical KL setting and clarify the relation between entropic regularization and information projection for $1 < q \leq 2$.
Comments24 pages
