Zhang Zhen, Wang Mianzhi, Nehorai Arye
IEEE Trans Pattern Anal Mach Intell. 2020 Jul;42(7):1741-1754. doi: 10.1109/TPAMI.2019.2903050. Epub 2019 Mar 4.
In this paper, we present a mathematical and computational framework for comparing and matching distributions in reproducing kernel Hilbert spaces (RKHS). This framework, called optimal transport in RKHS, is a generalization of the optimal transport problem in input spaces to (potentially) infinite-dimensional feature spaces. We provide a computable formulation of Kantorovich's optimal transport in RKHS. In particular, we explore the case in which data distributions in RKHS are Gaussian, obtaining closed-form expressions of both the estimated Wasserstein distance and optimal transport map via kernel matrices. Based on these expressions, we generalize the Bures metric on covariance matrices to infinite-dimensional settings, providing a new metric between covariance operators. Moreover, we extend the correlation alignment problem to Hilbert spaces, giving a new strategy for matching distributions in RKHS. Empirically, we apply the derived formulas under the Gaussianity assumption to image classification and domain adaptation. In both tasks, our algorithms yield state-of-the-art performances, demonstrating the effectiveness and potential of our framework.
在本文中，我们提出了一个用于在再生核希尔伯特空间（RKHS）中比较和匹配分布的数学与计算框架。这个框架，称为RKHS中的最优传输，是输入空间中最优传输问题到（潜在的）无限维特征空间的推广。我们给出了RKHS中康托罗维奇最优传输的可计算形式。特别地，我们探讨了RKHS中的数据分布为高斯分布的情况，通过核矩阵得到了估计的瓦瑟斯坦距离和最优传输映射的闭式表达式。基于这些表达式，我们将协方差矩阵上的布雷尔斯度量推广到无限维情形，给出了协方差算子之间的一种新度量。此外，我们将相关对齐问题扩展到希尔伯特空间，给出了一种在RKHS中匹配分布的新策略。从经验上看，我们在高斯性假设下将导出的公式应用于图像分类和域适应。在这两个任务中，我们的算法都取得了领先的性能，证明了我们框架的有效性和潜力。
