Sliced Multi-Marginal Optimal Transport
arXiv:2102.07115
Abstract
Multi-marginal optimal transport enables one to compare multiple probability measures, which increasingly finds application in multi-task learning problems. One practical limitation of multi-marginal transport is computational scalability in the number of measures, samples and dimensionality. In this work, we propose a multi-marginal optimal transport paradigm based on random one-dimensional projections, whose (generalized) distance we term the sliced multi-marginal Wasserstein distance. To construct this distance, we introduce a characterization of the one-dimensional multi-marginal Kantorovich problem and use it to highlight a number of properties of the sliced multi-marginal Wasserstein distance. In particular, we show that (i) the sliced multi-marginal Wasserstein distance is a (generalized) metric that induces the same topology as the standard Wasserstein distance, (ii) it admits a dimension-free sample complexity, (iii) it is tightly connected with the problem of barycentric averaging under the sliced-Wasserstein metric. We conclude by illustrating the sliced multi-marginal Wasserstein on multi-task density estimation and multi-dynamics reinforcement learning problems.
References in corpus (11)
- Fast Computation of Wasserstein Barycenters
- Gromov-Wasserstein Learning for Graph Matching and Node Embedding
- Sliced-Wasserstein Autoencoder: An Embarrassingly Simple Generative Model
- Primal Wasserstein Imitation Learning
- Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching
- Multi-marginal Wasserstein GAN
- On the Complexity of Approximating Multimarginal Optimal Transport
- Sliced Gromov-Wasserstein
- Statistical and Topological Properties of Sliced Probability Divergences
- Generalized Sliced Wasserstein Distances
- Estimating Barycenters of Measures in High Dimensions