Wasserstein distance, which measures the discrepancy between distributions, shows efficacy in various types of natural language processing (NLP) and computer vision (CV) applications. One of the challenges in estimating Wasserstein distance is that it is computationally expensive and does not scale well for many distribution comparison tasks. In this paper, we aim to approximate the 1-Wasserstein distance by the tree-Wasserstein distance (TWD), where TWD is a 1-Wasserstein distance with tree-based embedding and can be computed in linear time with respect to the number of nodes on a tree. More specifically, we propose a simple yet efficient L1-regularized approach to learning the weights of the edges in a tree. To this end, we first show that the 1-Wasserstein approximation problem can be formulated as a distance approximation problem using the shortest path distance on a tree. We then show that the shortest path distance can be represented by a linear model and can be formulated as a Lasso-based regression problem. Owing to the convex formulation, we can obtain a globally optimal solution efficiently. Moreover, we propose a tree-sliced variant of these methods. Through experiments, we demonstrated that the weighted TWD can accurately approximate the original 1-Wasserstein distance.
翻译:衡量分布差异的瓦瑟斯坦距离(Wasserstein距离)测量分布差异,显示了各种自然语言处理(NLP)和计算机视觉(CV)应用程序的功效。在估计瓦瑟斯坦距离(Wasserstein距离)方面,一个挑战在于它计算成本昂贵,对于许多分布比较任务来说规模不高。在本文中,我们的目标是通过树-Wasserstein距离(TWD),将Wasserstein距离约为1-Wasserstein距离(Wasserstein距离为1瓦瑟斯坦距离,与树基嵌植植植的距离为1瓦瑟斯坦距离),并且可以以线性时间计算一棵树上的节点数。更具体地说,我们提出了一个简单而高效的L1常规化方法来学习树边缘的重量。我们首先展示的是,1Wasserstein近似问题可以用树上最短的路径距离来形成。我们通过配置的矩形组合,可以有效地获得全球最佳解决方案。此外,我们还可以提出一种树-Wasserimal-WA iralalalalalalalal ad asildal destal 这样的方法。