Revisiting Tree-Sliced Wasserstein Distance Through the Lens of the Fermat–Weber Problem

Tree-Sliced methods have emerged as an efficient and expressive alternative to the traditional Sliced Wasserstein distance, replacing one-dimensional projections with tree-structured metric spaces and leveraging a splitting mechanism to better capture the underlying topological structure of integration domains while maintaining low computational cost. At the core of this framework is the Tree-Sliced Wasserstein (TSW) distance, defined over probability measures in Euclidean spaces, along with several variants designed to enhance its performance. A fundamental distinction between SW and TSW lies in their sampling strategies—a component explored in the context of SW but often overlooked in comparisons. This omission is significant: whereas SW relies exclusively on directional projections, TSW incorporates both directional and positional information through its tree-based construction. This enhanced spatial sensitivity enables TSW to reflect the geometric structure of the underlying data more accurately. Building on this insight, we propose a novel variant of TSW that explicitly leverages positional information in its design. Inspired by the classical Fermat–Weber problem—which seeks a point minimizing the sum of distances to a given set of points—we introduce the Fermat–Weber Tree-Sliced Wasserstein (FW-TSW) distance. By incorporating geometric median principles into the tree construction process, FW-TSW notably further improves the performance of TSW while preserving its low computational cost. These improvements are empirically validated across diverse experiments, including diffusion model training and gradient flow. Our code is available at https://github.com/thanhquangtran/FW-TSW.