We introduce a formulation of optimal transport problem for distributions on function spaces, where the stochastic map between functional domains can be partially represented in terms of an (infinite-dimensional) Hilbert-Schmidt operator mapping a Hilbert space of functions to another. For numerous machine learning tasks, data can be naturally viewed as samples drawn from spaces of functions, such as curves and surfaces, in high dimensions. Optimal transport for functional data analysis provides a useful framework of treatment for such domains. In this work, we develop an efficient algorithm for finding the stochastic transport map between functional domains and provide theoretical guarantees on the existence, uniqueness, and consistency of our estimate for the Hilbert-Schmidt operator. We validate our method on synthetic datasets and study the geometric properties of the transport map. Experiments on real-world datasets of robot arm trajectories further demonstrate the effectiveness of our method on applications in domain adaptation.
翻译:我们为功能空间的分布提出了最佳运输问题,其中功能领域之间的随机地图可以部分代表(无限)Hilbert-Schmidt操作员绘制Hilbert的功能空间图。对于许多机器学习任务,可以自然地将数据视为从功能空间(如曲线和表面)中提取的高维参数样本。功能数据分析的最佳运输为此类领域的处理提供了有用的框架。在这项工作中,我们开发了一种高效率的算法,用于查找功能领域之间的随机运输图,并就我们对Hilbert-Schmidt操作员的估计的存在、独特性和一致性提供理论保证。我们验证了我们合成数据集的方法,并研究了运输图的几何特性。对机器人臂轨迹的实际世界数据集的实验进一步证明了我们在领域适应应用方法的有效性。