使用条件归一化流计算最优传输映射和Wasserstein重心
Computing Optimal Transport Maps and Wasserstein Barycenters Using Conditional Normalizing Flows
AI总结：
本文提出一种利用条件归一化流高效计算高维最优传输映射和Wasserstein重心的新方法，通过基于梯度的传输成本最小化直接求解原始问题，突破了以往方法难以处理数百个输入分布的计算瓶颈。
摘要：
我们提出了一种在高维空间中高效计算最优传输映射和Wasserstein重心的全新方法。我们的方法使用条件归一化流将输入分布近似为从公共潜在空间出发的可逆前推变换。这使得通过基于梯度的传输成本最小化直接求解原始问题成为可能，而不同于以往依赖对偶形式和复杂对抗优化的方法。我们展示了如何通过求解条件方差最小化问题来扩展该方法以计算Wasserstein重心。我们条件架构的一个关键优势在于，它能够计算数百个输入分布的重心，而这在以往方法中是计算上不可行的。我们的数值实验表明，该方法在各种高维任务中均产生了准确的结果，并且与以往最先进的方法相比表现出色。
英文摘要：
We present a novel method for efficiently computing optimal transport maps and Wasserstein barycenters in high-dimensional spaces. Our approach uses conditional normalizing flows to approximate the input distributions as invertible pushforward transformations from a common latent space. This makes it possible to directly solve the primal problem using gradient-based minimization of the transport cost, unlike previous methods that rely on dual formulations and complex adversarial optimization. We show how this approach can be extended to compute Wasserstein barycenters by solving a conditional variance minimization problem. A key advantage of our conditional architecture is that it enables the computation of barycenters for hundreds of input distributions, which was computationally infeasible with previous methods. Our numerical experiments illustrate that our approach yields accurate results across various high-dimensional tasks and compares favorably with previous state-of-the-art methods.
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