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 at https://github.com/yixuan/temperflow.

References in corpus (21)

• Adam: A Method for Stochastic Optimization
• Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms
• MCMC using Hamiltonian dynamics
• Simulated Tempering: A New Monte Carlo Scheme
• NICE: Non-linear Independent Components Estimation
• Variational Inference with Normalizing Flows
• Parallel Tempering: Theory, Applications, and New Perspectives
• Dynamic temperature selection for parallel-tempering in Markov chain Monte Carlo simulations
• A Biased Monte Carlo Scheme for Zeolite Structure Solution
• Masked Autoregressive Flow for Density Estimation
• Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis
• Deep Knockoffs
• Roundtrip: A Deep Generative Neural Density Estimator
• Conditions for rapid mixing of parallel and simulated tempering on multimodal distributions
• NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport
• Your GAN is Secretly an Energy-based Model and You Should use Discriminator Driven Latent Sampling
• Exponential ergodicity of mirror-Langevin diffusions
• Simulated Tempering Langevin Monte Carlo II: An Improved Proof using Soft Markov Chain Decomposition
• Invertible Generative Modeling using Linear Rational Splines
• Schr{ö}dinger-F{ö}llmer Sampler: Sampling without Ergodicity
• Deep Generative Learning via Variational Gradient Flow