Sliced Wasserstein (SW) distance in Optimal Transport (OT) is widely used in various applications thanks to its statistical effectiveness and computational efficiency. On the other hand, Tree Wassenstein (TW) and Tree-sliced Wassenstein (TSW) are instances of OT for probability measures where its ground cost is a tree metric. TSW also has a low computational complexity, i.e. linear to the number of edges in the tree. Especially, TSW is identical to SW when the tree is a chain. While SW is prone to loss of topological information of input measures due to relying on one-dimensional projection, TSW is more flexible and has a higher degree of freedom by choosing a tree rather than a line to alleviate the curse of dimensionality in SW. However, for practical applications, popular tree metric sampling methods are heavily built upon given supports, which limits their capacity to adapt to new supports. In this paper, we propose the Tree-Sliced Wasserstein distance on a System of Lines (TSW-SL), which brings a connection between SW and TSW. Compared to SW and TSW, our TSW-SL benefits from the higher degree of freedom of TSW while being suitable to dynamic settings as SW. In TSW-SL, we use a variant of the Radon Transform to project measures onto a system of lines, resulting in measures on a space with a tree metric, then leverage TW to efficiently compute distances between them. We empirically verify the advantages of TSW-SL over the traditional SW by conducting a variety of experiments on gradient flows, image style transfer, and generative models.

翻译：最优传输中的切片瓦瑟斯坦距离因其统计有效性和计算高效性被广泛应用于各个领域。另一方面，树瓦瑟斯坦距离和树切片瓦瑟斯坦距离是最优传输中针对概率度量的特例，其基础成本为树度量。树切片瓦瑟斯坦距离同样具有较低的计算复杂度，即与树中边数呈线性关系。特别地，当树为链状结构时，树切片瓦瑟斯坦距离等同于切片瓦瑟斯坦距离。切片瓦瑟斯坦距离由于依赖一维投影容易损失输入度量的拓扑信息，而树切片瓦瑟斯坦距离通过选择树而非直线来缓解维度灾难，具有更高的灵活性和自由度。然而在实际应用中，常用的树度量采样方法严重依赖于给定支撑集，这限制了其适应新支撑集的能力。本文提出在线系统上的树切片瓦瑟斯坦距离，建立了切片瓦瑟斯坦距离与树切片瓦瑟斯坦距离之间的桥梁。相较于切片瓦瑟斯坦距离和树切片瓦瑟斯坦距离，我们的方法既保留了树切片瓦瑟斯坦距离的高自由度优势，又具备切片瓦瑟斯坦距离适用于动态场景的特性。该方法采用拉东变换的变体将度量投影到线系统上，从而获得树度量空间中的度量分布，进而利用树瓦瑟斯坦距离高效计算其间距。我们通过梯度流、图像风格迁移和生成模型等多组实验，实证验证了该方法相较于传统切片瓦瑟斯坦距离的优越性。