用条件归一化流高效计算高维空间的最优传输映射和巴氏中心。
Computing Optimal Transport Maps and Wasserstein Barycenters Using Conditional Normalizing Flows
- 基于条件归一化流，将分布映射到公共隐空间，直接优化运输成本。
- 可处理数百个输入分布的巴氏中心计算，显著提升效率。
- 适用于高维任务，结果优于现有最先进方法。
我们提出一种高效计算高维空间中最优传输映射和Wasserstein巴氏中心的新方法。该方法利用条件归一化流，将输入分布近似为从共同隐空间出发的可逆映射。这使得可以直接通过梯度下降最小化运输成本来求解原始问题，而无需依赖传统的对偶公式和复杂的对抗优化。我们进一步展示了如何通过求解条件方差最小化问题来计算巴氏中心。其关键优势在于，该条件架构支持对数百个输入分布计算巴氏中心，相较以往方法具有显著的计算可行性。数值实验表明，该方法在多种高维任务中均能获得精确结果，性能优于现有最先进方法。
原文摘要 · Abstract (English)
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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