An Optimal Transport Approach for the Schrödinger Bridge Problem and Convergence of Sinkhorn Algorithm
An Optimal Transport Approach for the Schrödinger Bridge Problem and Convergence of Sinkhorn Algorithm
复制标题
薛定谔桥问题的最优传输方法及Sinkhorn算法的收敛性
DOI：
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发表时间：
2019
影响因子：
2.5
通讯作者：
Augusto Gerolin
中科院分区：
文献类型：
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作者：
Simone Di Marino;Augusto Gerolin
本文利用了Schrödinger Bridge problem （l<s:2> in Funct Anal 262:1879-1920, 2012; Nelson in Phys Rev 150:1079, 1966; Schrödinger in Über die umkehrung der naturgesetze）之间的等价性。Verlag academie der wissenschaften委员会bei Walter de Gruyter， u, Company, 1931)和熵惩罚的最优运输(Cuturi：进展在神经信息处理系统，pp 2292-2300, 2013； Galichon和salani<s:1> in：匹配与权衡：揭示偏好的竞争特征。CEPR讨论文件第DP7858, 2010)，以寻找一种不同的方法来实现二元性，在最佳运输的精神。这种方法的结果是先验估计，当正则化参数趋于零时，先验估计在极限上是一致的。特别地，我们找到了一个新的证明，证明了最大熵势的存在性，从而证明了Schrödinger系统解的存在性。我们的方法在有两个以上的边缘情况下也得到了推广：主要的新结果是证明了Sinkhorn算法即使在连续的多边缘情况下也是收敛的。这也提供了Sinkhorn算法在两个边界上收敛的另一种证明。
This paper exploit the equivalence between the Schrödinger Bridge problem (Léonard in J Funct Anal 262:1879–1920, 2012; Nelson in Phys Rev 150:1079, 1966; Schrödinger in Über die umkehrung der naturgesetze. Verlag Akademie der wissenschaften in kommission bei Walter de Gruyter u, Company, 1931) and the entropy penalized optimal transport (Cuturi in: Advances in neural information processing systems, pp 2292–2300, 2013; Galichon and Salanié in: Matching with trade-offs: revealed preferences over competing characteristics. CEPR discussion paper no. DP7858, 2010) in order to find a different approach to the duality, in the spirit of optimal transport. This approach results in a priori estimates which are consistent in the limit when the regularization parameter goes to zero. In particular, we find a new proof of the existence of maximizing entropic-potentials and therefore, the existence of a solution of the Schrödinger system. Our method extends also when we have more than two marginals: the main new result is the proof that the Sinkhorn algorithm converges even in the continuous multi-marginal case. This provides also an alternative proof of the convergence of the Sinkhorn algorithm in two marginals.
影响因子：
1.9
作者：
Chen, Yongxin;Georgiou, Tryphon;Pavon, Michele
通讯作者：
Pavon, Michele