Amari Shun-Ichi, Karakida Ryo, Oizumi Masafumi, Cuturi Marco
RIKEN, Wako-shi, Saitama 351-0198, Japan
National Institute of Advanced Industrial Science and Technology, Koto-ku, Tokyo 135-0064, Japan
Neural Comput. 2019 May;31(5):827-848. doi: 10.1162/neco_a_01178. Epub 2019 Mar 18.
We propose a new divergence on the manifold of probability distributions, building on the entropic regularization of optimal transportation problems. As Cuturi ( 2013 ) showed, regularizing the optimal transport problem with an entropic term is known to bring several computational benefits. However, because of that regularization, the resulting approximation of the optimal transport cost does not define a proper distance or divergence between probability distributions. We recently tried to introduce a family of divergences connecting the Wasserstein distance and the Kullback-Leibler divergence from an information geometry point of view (see Amari, Karakida, & Oizumi, 2018 ). However, that proposal was not able to retain key intuitive aspects of the Wasserstein geometry, such as translation invariance, which plays a key role when used in the more general problem of computing optimal transport barycenters. The divergence we propose in this work is able to retain such properties and admits an intuitive interpretation.
我们基于最优传输问题的熵正则化，在概率分布流形上提出了一种新的散度。正如Cuturi（2013年）所表明的，用熵项对最优传输问题进行正则化已知会带来若干计算上的好处。然而，由于这种正则化，最优传输成本的所得近似并没有定义概率分布之间的恰当距离或散度。我们最近试图从信息几何的角度引入一族连接瓦瑟斯坦距离和库尔贝克 - 莱布勒散度的散度（见Amari、Karakida和Oizumi，2018年）。然而，该提议无法保留瓦瑟斯坦几何的关键直观方面，比如平移不变性，而平移不变性在用于计算最优传输重心的更一般问题时起着关键作用。我们在这项工作中提出的散度能够保留此类性质并允许一种直观的解释。
