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This paper presents a multiscale approach to efficiently compute approximateoptimal transport plans between point sets. It is particularly well-suited forpoint sets that are in high-dimensions, but are close to being intrinsicallylow-dimensional. The approach is based on an adaptive multiscale decompositionof the point sets. The multiscale decomposition yields a sequence of optimaltransport problems, that are solved in a top-to-bottom fashion from thecoarsest to the finest scale. We provide numerical evidence that thismultiscale approach scales approximately linearly, in time and memory, in thenumber of nodes, instead of quadratically or worse for a direct solution.Empirically, the multiscale approach results in less than one percent relativeerror in the objective function. Furthermore, the multiscale plans constructedare of interest by themselves as they may be used to introduce novel featuresand notions of distances between point sets. An analysis of sets of brain MRIbased on optimal transport distances illustrates the effectiveness of theproposed method on a real world data set. The application demonstrates thatmultiscale optimal transport distances have the potential to improve onstate-of-the-art metrics currently used in computational anatomy.

Multiscale Strategies for Computing Optimal Transport