BSP-OT: Sparse transport plans between discrete measures in loglinear time

Abstract

To solve the optimal transport problem between two uniform discrete measures of the same size, one seeks a bijective assignment that minimizes some matching cost. For this task, exact algorithms are intractable for large problems, while approximate ones may lose the bijectivity of the assignment. We address this issue and the more general cases of non-uniform discrete measures with different total masses, where partial transport may be desirable. The core of our algorithm is a variant of the Quicksort algorithm that provides an efficient strategy to randomly explore many relevant and easy-to-compute couplings, by matching BSP trees in loglinear time. The couplings we obtain are as sparse as possible, in the sense that they provide bijections, injective partial matchings or sparse couplings depending on the nature of the matched measures. To improve the transport cost, we propose efficient strategies to merge 𝑘 sparse couplings into a higher quality one. For 𝑘 = 64, we obtain transport plans with typically less than 1% of relative error in a matter of seconds between hundreds of thousands of points in 3D on the CPU. We demonstrate how these high-quality approximations can drastically speed-up usual pipelines involving optimal transport, such as shape interpolation, intrinsic manifold sampling, color transfer, topological data analysis, rigid partial registration of point clouds and image stippling.

To build a bijection between two point sets (here, stippled images of L. Kantorovich and G. Monge, left), we simultaneously build a pair of BSP-tree on both sets, by recursively splitting the points along the same random direction. We then match the corresponding leaves of the trees to obtain an assignation. After building such 𝑘 assignations in parallel, we combine them into a single one, of low transport cost (color-coded on the right). This allows for numerous applications in computer graphics.