Projection Optimal Transport on Tree-Ordered Lines

Hoang V. Tran, Huyen Trang Pham, Tho Tran Huu, Minh-Khoi Nguyen-Nhat, Thanh Chu, Tam Le, Tan Minh Nguyen

Keywords: projection optimal transport, optimal transport

TL;DR: In this paper, we replace one-dimensional lines in Sliced Optimal Transport with a more intricate structure, called tree systems, where OT problems on tree systems still allow closed-form solutions

Abstract: 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 \emph{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.

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

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Submission Number: 7837