On the difference between entropic cost and the optimal transport cost
On the difference between entropic cost and the optimal transport cost
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论熵成本与最优运输成本的区别
DOI：
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Soumik Pal
中科院分区：
文献类型：
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作者：
Soumik Pal
考虑使用严格凸代价函数在$mathbb{R}^d$上传输密度$
ho_0$到$
ho_1$的monage - kantorovich问题。这个问题的一种流行的松弛是称为熵成本问题的单参数族。熵成本$K_h$, $h>0$的计算速度要快得多，并且当$h$趋于零时，$h K_h$收敛于最优运输成本。我们感兴趣的是收敛速度。我们展示了$K_h$和$1/h$之间的差值乘以最优运输成本在将紧凑支持的密度运输到满足其他一些技术限制的另一个密度时具有点向限制。这个极限是$
ho_1$相对于$mathbb{R}^d$上的黎曼体积度量的相对熵，这个黎曼体积度量测量传输图的局部灵敏度。对于二次Wasserstein输运，这个相对熵正好是$
ho_1$和$
ho_0$熵差的一半。在这种情况下，我们补充了Adams等人，Duong等人和Erbar等人的结果，他们都使用伽玛收敛。更令人惊讶的是，我们证明了两个熵的差异（加上成本）也是Pal和Wong最近引入的狄利克雷输运的极限。后者可以被认为是沃瑟斯坦输运的乘法模拟，对应于非局部算子。从Jordan-Kinderlehrer-Otto的意义上说，它暗示了熵的潜在梯度流，即使成本函数不是度量。这些证明是基于薛定谔桥在$h$趋近于零时的高斯近似。
Consider the Monge-Kantorovich problem of transporting densities $
ho_0$ to $
ho_1$ on $mathbb{R}^d$ with a strictly convex cost function. A popular relaxation of the problem is the one-parameter family called the entropic cost problem. The entropic cost $K_h$, $h>0$, is significantly faster to compute and $h K_h$ is known to converge to the optimal transport cost as $h$ goes to zero. We are interested the rate of convergence. We show that the difference between $K_h$ and $1/h$ times the optimal cost of transport has a pointwise limit when transporting a compactly supported density to another that satisfies a few other technical restrictions. This limit is the relative entropy of $
ho_1$ with respect to a Riemannian volume measure on $mathbb{R}^d$ that measures the local sensitivity of the transport map. For the quadratic Wasserstein transport, this relative entropy is exactly one half of the difference of entropies of $
ho_1$ and $
ho_0$. In that case we complement the results of Adams et al., Duong et al, and Erbar et al. who all use gamma convergence. More surprisingly, we demonstrate that this difference of two entropies (plus the cost) is also the limit for the Dirichlet transport introduced recently by Pal and Wong. The latter can be thought of as a multiplicative analog of the Wasserstein transport and corresponds to a non-local operator. It hints at an underlying gradient flow of entropy, in the sense of Jordan-Kinderlehrer-Otto, even when the cost function is not a metric. The proofs are based on Gaussian approximations to Schrodinger bridges as $h$ approaches zero.
影响因子：
2.5
作者：
Bernton, Espen;Ghosal, Promit;Nutz, Marcel
通讯作者：
Nutz, Marcel