2025年度 統計数理セミナー
Statistical Mathematics Seminar 2025

English

日本語

日本語

日本語

日本語

日本語

日本語

日本語

日本語

日本語

English

Optimal transport (OT) provides a set of powerful toolkits to compare measures. However, OT has a high computational complexity, i.e., super cubic w.r.t. the number of input supports. Several variants of Sliced Wasserstein (SW) have been developed in the literature to overcome this challenge. These approaches exploit the closed-form expression of the univariate OT by projecting input measures onto one-dimensional lines. However, projecting measures onto low-dimensional spaces can lead to a loss of topological information. To mitigate this issue, we propose to replace one-dimensional lines with a more advanced structure, called tree systems. This structure is metrizable by a tree metric, which yields a closed-form expression for OT on tree systems. We derive an extensive theoretical analysis to formally define tree systems, introduce the concept of splitting maps, propose novel variants of Radon transform for tree systems, and verify their infectivity. Empirically, we illustrate that the proposed approaches perform favorably compared to SW and its variants on applications with dynamic-support measures such as generative models, and diffusion models.

日本語

日本語

日本語