Sampling from high-dimensional distributions is a fundamental problem in statistical research and practice. However, great challenges emerge when the target density function is unnormalized and contains isolated modes. We tackle this difficulty by fitting an invertible transformation mapping, called a transport map, between a reference probability measure and the target distribution, so that sampling from the target distribution can be achieved by pushing forward a reference sample through the transport map. We theoretically analyze the limitations of existing transport-based sampling methods using the Wasserstein gradient flow theory, and propose a new method called TemperFlow that addresses the multimodality issue. TemperFlow adaptively learns a sequence of tempered distributions to progressively approach the target distribution, and we prove that it overcomes the limitations of existing methods. Various experiments demonstrate the superior performance of this novel sampler compared to traditional methods, and we show its applications in modern deep learning tasks such as image generation. The programming code for the numerical experiments is available at https://github.com/yixuan/temperflow.

翻译：从高维分布中进行采样是统计研究与实践中的基本问题。然而，当目标密度函数未归一化且包含孤立模式时，会面临巨大挑战。我们通过拟合一个可逆变换映射（称为传输映射），在参考概率测度与目标分布之间建立关联，从而通过将参考样本经传输映射向前推演来实现对目标分布的采样。基于Wasserstein梯度流理论，我们从理论上分析了现有基于传输的采样方法的局限性，并提出了一种名为TemperFlow的新方法，以解决多模态问题。TemperFlow自适应地学习一系列调温分布，逐步逼近目标分布，我们证明了该方法克服了现有方法的局限性。多项实验表明，与经典方法相比，这种新型采样器具有更优性能，并展示了其在图像生成等现代深度学习任务中的应用。数值实验的编程代码可在https://github.com/yixuan/temperflow获取。