Score-based Generative Neural Networks for Large-Scale Optimal Transport

• URL: http://arxiv.org/abs/2110.03237v1
• Date: Thu, 7 Oct 2021 07:45:39 GMT
• Title: Score-based Generative Neural Networks for Large-Scale Optimal Transport
• Authors: Max Daniels, Tyler Maunu, Paul Hand
• Abstract summary: In certain cases, the optimal transport plan takes the form of a one-to-one mapping from the source support to the target support. We study instead the Sinkhorn problem, a regularized form of optimal transport whose solutions are couplings between the source and the target distribution. We introduce a novel framework for learning the Sinkhorn coupling between two distributions in the form of a score-based generative model.
• Score: 15.666205208594565
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the fundamental problem of sampling the optimal transport coupling between given source and target distributions. In certain cases, the optimal transport plan takes the form of a one-to-one mapping from the source support to the target support, but learning or even approximating such a map is computationally challenging for large and high-dimensional datasets due to the high cost of linear programming routines and an intrinsic curse of dimensionality. We study instead the Sinkhorn problem, a regularized form of optimal transport whose solutions are couplings between the source and the target distribution. We introduce a novel framework for learning the Sinkhorn coupling between two distributions in the form of a score-based generative model. Conditioned on source data, our procedure iterates Langevin Dynamics to sample target data according to the regularized optimal coupling. Key to this approach is a neural network parametrization of the Sinkhorn problem, and we prove convergence of gradient descent with respect to network parameters in this formulation. We demonstrate its empirical success on a variety of large scale optimal transport tasks.

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