论文标题
一般熵最佳运输成本的融合率
Convergence rate of general entropic optimal transport costs
论文作者
论文摘要
我们将最佳熵成本的收敛速率$ v_ \ varepsilon $作为噪声参数$ \ varepsilon \ downarrow 0 $。 We show that for a large class of cost functions $c$ on $\mathbb{R}^d\times \mathbb{R}^d$ (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and $L^{\infty}$ marginals, one has $v_\varepsilon-v_0= \frac{d}{2} \ varepsilon \ log（1/\ varepsilon）+ o（\ varepsilon）$。上限是通过块近似策略和Alexandrov定理的整体变体获得的。在$ c $的无限扭曲条件下，即$ \ nabla_ {xy}^2 c（x，y）$的可逆性，我们通过建立$ d $ dimensions duality差距的二次分离来获得下限，这要归功于minty的技巧。
We investigate the convergence rate of the optimal entropic cost $v_\varepsilon$ to the optimal transport cost as the noise parameter $\varepsilon \downarrow 0$. We show that for a large class of cost functions $c$ on $\mathbb{R}^d\times \mathbb{R}^d$ (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and $L^{\infty}$ marginals, one has $v_\varepsilon-v_0= \frac{d}{2} \varepsilon \log(1/\varepsilon)+ O(\varepsilon)$. Upper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov's theorem. Under an infinitesimal twist condition on $c$, i.e. invertibility of $\nabla_{xy}^2 c(x,y)$, we get the lower bound by establishing a quadratic detachment of the duality gap in $d$ dimensions thanks to Minty's trick.
