Sliced Multi-Marginal Optimal Transport
This addresses computational bottlenecks for researchers and practitioners in multi-task learning, though it is incremental as it builds on existing sliced-Wasserstein methods.
The paper tackles the computational scalability issue in multi-marginal optimal transport by proposing a sliced multi-marginal Wasserstein distance based on random one-dimensional projections, showing it is a metric with dimension-free sample complexity and applying it to multi-task density estimation and reinforcement learning.
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.