Efficient Multimodal Sampling via Tempered Distribution Flow

Abstract

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 in the supplementary material.

keyword:

• Deep neural network
• Gradient flow
• Markov chain Monte Carlo
• Normalizing flow
• Parallel tempering

Supplementary Materials

The supplementary materials include more background information of the proposed method, additional experiment details, and the proofs of theorems in the main article.

Acknowledgments

The authors would like to thank the editor, the associate editor, and four reviewers for their insightful and constructive comments that have greatly helped improve this article.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

Yixuan Qiu’s work was supported in part by National Natural Science Foundation of China (12101389), Shanghai Pujiang Program (21PJC056), MOE Project of Key Research Institute of Humanities and Social Sciences (22JJD110001), and Shanghai Research Center for Data Science and Decision Technology.