Stochastic Normalizing Flows

By Hao Wu, Jonas Köhler, Frank No'e. 2020

224 citations recorded by the source corpus.

The sampling of probability distributions specified up to a normalization constant is an important problem in both machine learning and statistical mechanics. While classical stochastic sampling methods such as Markov Chain Monte Carlo (MCMC) or Langevin Dynamics (LD) can suffer from slow mixing times there is a growing interest in using normalizing flows in order to learn the transformation of a simple prior distribution to the given target distribution. Here we propose a generalized and combined approach to sample target densities: Stochastic Normalizing Flows (SNF) -- an arbitrary sequence of deterministic invertible functions and stochastic sampling blocks. We show that stochasticity overcomes expressivity limitations of normalizing flows resulting from the invertibility constraint, whereas trainable transformations between sampling steps improve efficiency of pure MCMC/LD along the flow. By invoking ideas from non-equilibrium statistical mechanics we derive an efficient training procedure by which both the sampler's and the flow's parameters can be optimized end-to-end, and by which we can compute exact importance weights without having to marginalize out the randomness of the stochastic blocks. We illustrate the representational power, sampling efficiency and asymptotic correctness of SNFs on several benchmarks including applications to sampling molecular systems in equilibrium.

Subfields: Management Science and Operations Research, Ocean Engineering, Statistics and Probability, Generative Models

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Citations in the corpus

Denoising Diffusion Probabilistic Models (2020, methodology): “Other methods for learning transition operators of Markov chains include infusion training [2], variational walkback [15], generative stochastic networks [1], and others [50, 54, 36, 42, 35, 65].”
Diffusion Normalizing Flow (2021, methodology): “As pointed out in Stochastic Normalizing Flows (SNF) [42], stochasticity can effectively improve the expressive power of the flow-based model in low dimension applications.”
Neural Controlled Differential Equations for Irregular Time Series (2020, methodology): “Adjoint backpropagationneeds some work to apply to SDEs, and so [42, 43, 44] all propose methods for training Neural SDEs.”
Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models (2021, methodology): “flows by interleaving small energy models, sampled with MCMC, between bijectors [244].”
The Power Spherical distribution (2020, background): “…De Cao et al., 2019) have been shown to be remarkably ﬂexible, they are still subject to topological constraints (Dinh et al., 2019; Cor-nish et al., 2019), which motivates extensions for complex manifolds (Brehmer & Cranmer, 2020; Wu et al., 2020) including spheres and tori (Rezende et al., 2020).”
Diffusion Models: A Comprehensive Survey of Methods and Applications (2022, background): “Unfortunately, the bijective requirement also has limitations on modeling complex data, both empirically and theoretically [39, 243].”