论文标题
局部最佳运输，并应用于积极的未标记学习
Partial Optimal Transport with Applications on Positive-Unlabeled Learning
论文作者
论文摘要
经典的最佳运输问题寻求一张运输图，该图保留了两个概率分布的总质量，要求其质量相同。在某些应用（例如颜色或形状匹配）中，这可能过于限制，因为分布可能具有任意质量和/或只有一小部分总质量的质量。已经设计了几种算法，用于计算依赖熵正则化的部分Wasserstein指标，但是当它带有确切的解决方案时，几乎没有沃斯史坦（Wasserstein）和Gromov-Wasserstein的部分表述。这排除了不在相同的度量空间或需要旋转或翻译不变时的分布。在本文中，我们解决了部分Wasserstein和Gromov-Wasserstein问题，并提出了确切的算法来解决它们。我们在一个积极的（PU）学习应用中展示了新的配方。据我们所知，这是在这种情况下的最佳运输的第一个应用，我们首先强调了部分基于Wasserstein的指标在通常的PU学习环境中有效。然后，我们证明部分Gromov-Wasserstein指标在点云来自不同域或具有不同特征的情况下有效。
Classical optimal transport problem seeks a transportation map that preserves the total mass betwenn two probability distributions, requiring their mass to be the same. This may be too restrictive in certain applications such as color or shape matching, since the distributions may have arbitrary masses and/or that only a fraction of the total mass has to be transported. Several algorithms have been devised for computing partial Wasserstein metrics that rely on an entropic regularization, but when it comes with exact solutions, almost no partial formulation of neither Wasserstein nor Gromov-Wasserstein are available yet. This precludes from working with distributions that do not lie in the same metric space or when invariance to rotation or translation is needed. In this paper, we address the partial Wasserstein and Gromov-Wasserstein problems and propose exact algorithms to solve them. We showcase the new formulation in a positive-unlabeled (PU) learning application. To the best of our knowledge, this is the first application of optimal transport in this context and we first highlight that partial Wasserstein-based metrics prove effective in usual PU learning settings. We then demonstrate that partial Gromov-Wasserstein metrics is efficient in scenario where point clouds come from different domains or have different features.
