Score-based Generative Neural Networks for Large-Scale Optimal Transport
Score-based Generative Neural Networks for Large-Scale Optimal Transport
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发表时间：
2021-10
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通讯作者：
Max Daniels;Tyler Maunu;Paul Hand
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作者：
Max Daniels;Tyler Maunu;Paul Hand
我们考虑对给定源分布和目标分布之间的最佳传输耦合进行采样的基本问题。在某些情况下，最佳传输计划采用从源支持到目标支持的一对一映射的形式，但由于线性编程例程的高成本和固有的维数灾难，对于大型和高维数据集来说，学习甚至近似这样的映射在计算上具有挑战性。相反，我们研究 Sinkhorn 问题，这是一种正则化的最优传输形式，其解决方案是源分布和目标分布之间的耦合。我们引入了一种新颖的框架，用于以基于分数的生成模型的形式学习两个分布之间的 Sinkhorn 耦合。以源数据为条件，我们的程序迭代 Langevin Dynamics，根据正则化最优耦合对目标数据进行采样。这种方法的关键是 Sinkhorn 问题的神经网络参数化，我们证明了梯度下降相对于该公式中的网络参数的收敛性。我们展示了其在各种大规模最优运输任务上的实证成功。
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.