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.

本文提出了一种称为TemperFlow的新方法，通过学习一系列调和分布来逐步接近目标分布，通过传输映射从高维度分布中采样，解决了因目标密度函数不规范且包含孤立模式而产生的巨大挑战，该方法得到了实验的证明和应用。

高效多模态采样：通过绝缘分布流