Tree-Sliced Wasserstein Distance: A Geometric Perspective
This work addresses the computational and topological limitations of Sliced Wasserstein distance for researchers and practitioners in machine learning, offering an incremental improvement with a novel geometric perspective.
The authors tackled the computational burden of Optimal Transport by proposing Tree-Sliced Wasserstein distance, which uses tree systems instead of one-dimensional lines to preserve more topological information, and demonstrated favorable performance in experiments on gradient flows, image style transfer, and generative models compared to existing methods.
Many variants of Optimal Transport (OT) have been developed to address its heavy computation. Among them, notably, Sliced Wasserstein (SW) is widely used for application domains by projecting the OT problem onto one-dimensional lines, and leveraging the closed-form expression of the univariate OT to reduce the computational burden. However, projecting measures onto low-dimensional spaces can lead to a loss of topological information. To mitigate this issue, in this work, we propose to replace one-dimensional lines with a more intricate structure, called tree systems. This structure is metrizable by a tree metric, which yields a closed-form expression for OT problems on tree systems. We provide an extensive theoretical analysis to formally define tree systems with their topological properties, introduce the concept of splitting maps, which operate as the projection mechanism onto these structures, then finally propose a novel variant of Radon transform for tree systems and verify its injectivity. This framework leads to an efficient metric between measures, termed Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL). By conducting a variety of experiments on gradient flows, image style transfer, and generative models, we illustrate that our proposed approach performs favorably compared to SW and its variants.