In many fields of science, high-dimensional integration is required. Numerical methods have been developed to evaluate these complex integrals. We introduce the code i-flow, a python package that performs high-dimensional numerical integration utilizing normalizing flows. Normalizing flows are machine-learned, bijective mappings between two distributions. i-flow can also be used to sample random points according to complicated distributions in high dimensions. We compare i-flow to other algorithms for high-dimensional numerical integration and show that i-flow outperforms them for high dimensional correlated integrals. The i-flow code is publicly available on gitlab at https://gitlab.com/i-flow/i-flow.
Department of Electrical and Computer Engineering（电气与计算机工程系）；Duke University（杜克大学）
We introduce Projected Latent Markov Chain Monte Carlo (PL-MCMC), a technique for sampling from the high-dimensional conditional distributions learned by a normalizing flow. We prove that a Metropolis-Hastings implementation of PL-MCMC asymptotically samples from the exact conditional distributions associated with a normalizing flow. As a conditional sampling method, PL-MCMC enables Monte Carlo Expectation Maximization (MC-EM) training of normalizing flows from incomplete data. Through experimental tests applying normalizing flows to missing data tasks for a variety of data sets, we demonstrat…
Peking University（北京大学）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Lab（劳伦斯伯克利国家实验室）
Normalizing constant (also called partition function, Bayesian evidence, or marginal likelihood) is one of the central goals of Bayesian inference, yet most of the existing methods are both expensive and inaccurate. Here we develop a new approach, starting from posterior samples obtained with a standard Markov Chain Monte Carlo (MCMC). We apply a novel Normalizing Flow (NF) approach to obtain an analytic density estimator from these samples, followed by Optimal Bridge Sampling (OBS) to obtain the normalizing constant. We compare our method which we call Gaussianized Bridge Sampling (GBS) to ex…
Institut de Física d´Altes Energies (IFAE) - Barcelona Institute of Science and Technology (BIST)（高能物理研究所（IFAE）-巴塞罗那科学技术研究院（BIST））；Aplicaciones en Informática Avanzada (AIA)（高级信息技术应用公司（AIA））；University of Geneva（日内瓦大学）
For many applications, such as computing the expected value of different magnitudes, sampling from a known probability density function, the target density, is crucial but challenging through the inverse transform. In these cases, rejection and importance sampling require suitable proposal densities, which can be evaluated and sampled from efficiently. We will present a method based on normalizing flows, proposing a solution for the common problem of exploding reverse Kullback-Leibler divergence due to the target density having values of 0 in regions of the flow transformation. The performance…
Huawei R&D UK（华为英国研发中心）；University College London（伦敦大学学院）
We present a method for conditional sampling for pre-trained normalizing flows when only part of an observation is available. We derive a lower bound to the conditioning variable log-probability using Schur complement properties in the spirit of Gaussian conditional sampling. Our derivation relies on partitioning flow's domain in such a way that the flow restrictions to subdomains remain bijective, which is crucial for the Schur complement application. Simulation from the variational conditional flow then amends to solving an equality constraint. Our contribution is three-fold: a) we provide d…
Flatiron Institute（熨斗研究所）；Center for Data Science, New York University（纽约大学数据科学中心）；Stanford University（斯坦福大学）；Courant Institute, New York University（纽约大学柯朗研究所）
Normalizing flows can generate complex target distributions and thus show promise in many applications in Bayesian statistics as an alternative or complement to MCMC for sampling posteriors. Since no data set from the target posterior distribution is available beforehand, the flow is typically trained using the reverse Kullback-Leibler (KL) divergence that only requires samples from a base distribution. This strategy may perform poorly when the posterior is complicated and hard to sample with an untrained normalizing flow. Here we explore a distinct training strategy, using the direct KL diver…
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 …
To overcome topological constraints and improve the expressiveness of normalizing flow architectures, Wu, Köhler and Noé introduced stochastic normalizing flows which combine deterministic, learnable flow transformations with stochastic sampling methods. In this paper, we consider stochastic normalizing flows from a Markov chain point of view. In particular, we replace transition densities by general Markov kernels and establish proofs via Radon-Nikodym derivatives which allows to incorporate distributions without densities in a sound way. Further, we generalize the results for sampling from p…
International Institute of Information Technology, Hyderabad（海得拉巴国际信息技术学院）
Invertible convolutions have been an essential element for building expressive normalizing flow-based generative models since their introduction in Glow. Several attempts have been made to design invertible $k \times k$ convolutions that are efficient in training and sampling passes. Though these attempts have improved the expressivity and sampling efficiency, they severely lagged behind Glow which used only $1 \times 1$ convolutions in terms of sampling time. Also, many of the approaches mask a large number of parameters of the underlying convolution, resulting in lower expressivity on a fixe…
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所（HIIT））；University of Helsinki（赫尔辛基大学）
Variational approximations are increasingly based on gradient-based optimization of expectations estimated by sampling. Handling discrete latent variables is then challenging because the sampling process is not differentiable. Continuous relaxations, such as the Gumbel-Softmax for categorical distribution, enable gradient-based optimization, but do not define a valid probability mass for discrete observations. In practice, selecting the amount of relaxation is difficult and one needs to optimize an objective that does not align with the desired one, causing problems especially with models havi…
Amortized Sampling with Transferable Normalizing Flows
University of Oxford（牛津大学）；Université de Montréal（蒙特利尔大学）；Mila - Quebec AI Institute（魁北克AI研究所）；Freie Universität Berlin（柏林自由大学）；Valence Labs（Valence实验室）；AITHYRA；Institut Courtois（科特伊研究所）
Normalizing Flows (NFs) are emerging as a powerful class of generative models, as they not only allow for efficient sampling, but also deliver, by construction, density estimation. They are of great potential usage in High Energy Physics (HEP), where complex high dimensional data and probability distributions are everyday's meal. However, in order to fully leverage the potential of NFs it is crucial to explore their robustness as data dimensionality increases. Thus, in this contribution, we discuss the performances of some of the most popular types of NFs on the market, on some toy data sets w…
Normalizing Flows are generative models which produce tractable distributions where both sampling and density evaluation can be efficient and exact. The goal of this survey article is to give a coherent and comprehensive review of the literature around the construction and use of Normalizing Flows for distribution learning. We aim to provide context and explanation of the models, review current state-of-the-art literature, and identify open questions and promising future directions.
Recently proposed normalizing flow models such as Glow have been shown to be able to generate high quality, high dimensional images with relatively fast sampling speed. Due to their inherently restrictive architecture, however, it is necessary that they are excessively deep in order to train effectively. In this paper we propose to combine Glow with an underlying variational autoencoder in order to counteract this issue. We demonstrate that our proposed model is competitive with Glow in terms of image quality and test likelihood while requiring far less time for training.
Detecting and Mitigating Mode-Collapse for Flow-based Sampling of Lattice Field Theories
University of Bonn（波恩大学）；Helmholtz Institute for Radiation and Nuclear Physics (HISKP)（亥姆霍兹辐射与核物理研究所）；Berlin Institute for the Foundations of Learning and Data (BIFOLD)（柏林学习与数据基础研究所）；Technische Universität Berlin（柏林工业大学）；Northeastern University - London（东北大学伦敦校区）；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器研究所）；Roche（罗氏公司）；RIKEN Center for AIP（理化学研究所先进智能研究中心）
In this work, we propose a new family of generative flows on an augmented data space, with an aim to improve expressivity without drastically increasing the computational cost of sampling and evaluation of a lower bound on the likelihood. Theoretically, we prove the proposed flow can approximate a Hamiltonian ODE as a universal transport map. Empirically, we demonstrate state-of-the-art performance on standard benchmarks of flow-based generative modeling.
Fast inference of numerical model parameters from data is an important prerequisite to generate predictive models for a wide range of applications. Use of sampling-based approaches such as Markov chain Monte Carlo may become intractable when each likelihood evaluation is computationally expensive. New approaches combining variational inference with normalizing flow are characterized by a computational cost that grows only linearly with the dimensionality of the latent variable space, and rely on gradient-based optimization instead of sampling, providing a more efficient approach for Bayesian i…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Aurora Innovation（奥罗拉创新公司）
A popular way to plan trajectories in dynamic urban scenarios for Autonomous Vehicles is to rely on explicitly specified and hand crafted cost functions, coupled with random sampling in the trajectory space to find the minimum cost trajectory. Such methods require a high number of samples to find a low-cost trajectory and might end up with a highly suboptimal trajectory given the planning time budget. We explore the application of normalizing flows for improving the performance of trajectory planning for autonomous vehicles (AVs). Our key insight is to learn a sampling policy in a low-dimensio…
FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formula
IMT Atlantique, Lab-STICC, UMR 6285（IMT Atlantique，Lab-STICC，UMR 6285）；Univ Brest, CNRS, UMR 6205（Univ Brest，CNRS，UMR 6205）；Laboratoire de Mathématiques de Bretagne Atlantique（Bretagne Atlantique 数学实验室）
We consider the closely related problems of sampling from a distribution known up to a normalizing constant, and estimating said normalizing constant. We show how variational autoencoders (VAEs) can be applied to this task. In their standard applications, VAEs are trained to fit data drawn from an intractable distribution. We invert the logic and train the VAE to fit a simple and tractable distribution, on the assumption of a complex and intractable latent distribution, specified up to normalization. This procedure constructs approximations without the use of training data or Markov chain Mont…
Normalizing flows for lattice gauge theory in arbitrary space-time dimension
Massachusetts Institute of Technology（麻省理工学院）；New York University（纽约大学）；DeepMind（深度思维）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of Bern（伯尔尼大学）
Tsinghua University（清华大学）；University of California（加利福尼亚大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We propose a general purpose Bayesian inference algorithm for expensive likelihoods, replacing the stochastic term in the Langevin equation with a deterministic density gradient term. The particle density is evaluated from the current particle positions using a Normalizing Flow (NF), which is differentiable and has good generalization properties in high dimensions. We take advantage of NF preconditioning and NF based Metropolis-Hastings updates for a faster convergence. We show on various examples that the method is competitive against state of the art sampling methods.
Statistical procedures such as Bayes factor model selection and Bayesian model averaging require the computation of normalizing constants (e.g., marginal likelihoods). These normalizing constants are notoriously difficult to obtain, as they usually involve high-dimensional integrals that cannot be solved analytically. Here we introduce an R package that uses bridge sampling (Meng & Wong, 1996; Meng & Schilling, 2002) to estimate normalizing constants in a generic and easy-to-use fashion. For models implemented in Stan, the estimation procedure is automatic. We illustrate the functionality of t…
Verisk Analytics（Verisk分析公司）；University of Maryland, College Park（马里兰大学帕克分校）；Motive Technologies, Inc.（Motive科技公司）
Normalizing flows provide an elegant approach to generative modeling that allows for efficient sampling and exact density evaluation of unknown data distributions. However, current techniques have significant limitations in their expressivity when the data distribution is supported on a low-dimensional manifold or has a non-trivial topology. We introduce a novel statistical framework for learning a mixture of local normalizing flows as "chart maps" over the data manifold. Our framework augments the expressivity of recent approaches while preserving the signature property of normalizing flows, …
Microsoft Research AI（微软研究院人工智能部门）；University of Pittsburgh（匹兹堡大学）
The Normalizing Flow (NF) models a general probability density by estimating an invertible transformation applied on samples drawn from a known distribution. We introduce a new type of NF, called Deep Diffeomorphic Normalizing Flow (DDNF). A diffeomorphic flow is an invertible function where both the function and its inverse are smooth. We construct the flow using an ordinary differential equation (ODE) governed by a time-varying smooth vector field. We use a neural network to parametrize the smooth vector field and a recursive neural network (RNN) for approximating the solution of the ODE. Ea…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of Cambridge（剑桥大学）
Normalizing flows are a popular class of models for approximating probability distributions. However, their invertible nature limits their ability to model target distributions whose support have a complex topological structure, such as Boltzmann distributions. Several procedures have been proposed to solve this problem but many of them sacrifice invertibility and, thereby, tractability of the log-likelihood as well as other desirable properties. To address these limitations, we introduce a base distribution for normalizing flows based on learned rejection sampling, allowing the resulting norm…
We present a novel generative modeling method called diffusion normalizing flow based on stochastic differential equations (SDEs). The algorithm consists of two neural SDEs: a forward SDE that gradually adds noise to the data to transform the data into Gaussian random noise, and a backward SDE that gradually removes the noise to sample from the data distribution. By jointly training the two neural SDEs to minimize a common cost function that quantifies the difference between the two, the backward SDE converges to a diffusion process the starts with a Gaussian distribution and ends with the des…
Efficient gradient computation of the Jacobian determinant term is a core problem in many machine learning settings, and especially so in the normalizing flow framework. Most proposed flow models therefore either restrict to a function class with easy evaluation of the Jacobian determinant, or an efficient estimator thereof. However, these restrictions limit the performance of such density models, frequently requiring significant depth to reach desired performance levels. In this work, we propose Self Normalizing Flows, a flexible framework for training normalizing flows by replacing expensive…
We study a normalizing flow in the latent space of a top-down generator model, in which the normalizing flow model plays the role of the informative prior model of the generator. We propose to jointly learn the latent space normalizing flow prior model and the top-down generator model by a Markov chain Monte Carlo (MCMC)-based maximum likelihood algorithm, where a short-run Langevin sampling from the intractable posterior distribution is performed to infer the latent variables for each observed example, so that the parameters of the normalizing flow prior and the generator can be updated with …
Normalizing Flows are a promising new class of algorithms for unsupervised learning based on maximum likelihood optimization with change of variables. They offer to learn a factorized component representation for complex nonlinear data and, simultaneously, yield a density function that can evaluate likelihoods and generate samples. Despite these diverse offerings, applications of Normalizing Flows have focused primarily on sampling and likelihoods, with little emphasis placed on feature representation. A lack of theoretical foundation has left many open questions about how to interpret and app…
Simulating the Hubbard Model with Equivariant Normalizing Flows
University of Bonn（波恩大学）；Institute for Advanced Simulation 4 (IAS-4)（高级仿真研究所4）；Center for Advanced Simulation and Analytics (CASA)（高级仿真与分析中心）；University of the Virgin Islands（美属维尔京群岛大学）；College of Science and Mathematics（科学与数学学院）
Uncertainty quantification provides quantitative measures on the reliability of candidate solutions of ill-posed inverse problems. Due to their sequential nature, Monte Carlo sampling methods require large numbers of sampling steps for accurate Bayesian inference and are often computationally infeasible for large-scale inverse problems, such as seismic imaging. Our main contribution is a data-driven variational inference approach where we train a normalizing flow (NF), a type of invertible neural net, capable of cheaply sampling the posterior distribution given previously unseen seismic data f…
By chaining a sequence of differentiable invertible transformations, normalizing flows (NF) provide an expressive method of posterior approximation, exact density evaluation, and sampling. The trend in normalizing flow literature has been to devise deeper, more complex transformations to achieve greater flexibility. We propose an alternative: Gradient Boosted Normalizing Flows (GBNF) model a density by successively adding new NF components with gradient boosting. Under the boosting framework, each new NF component optimizes a sample weighted likelihood objective, resulting in new components th…
Efficient Regression-Based Training of Normalizing Flows for Boltzmann Generators
Mila – Québec AI Institute（魁北克AI研究所）；Université de Montréal（蒙特利尔大学）；Massachusetts Institute of Technology（麻省理工学院）；University of Oxford（牛津大学）；AITHYRA；HEC Montréal（蒙特利尔HEC）
Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）；McGill University（麦吉尔大学）；Facebook AI Research（脸书人工智能研究院）
Randomized value functions offer a promising approach towards the challenge of efficient exploration in complex environments with high dimensional state and action spaces. Unlike traditional point estimate methods, randomized value functions maintain a posterior distribution over action-space values. This prevents the agent's behavior policy from prematurely exploiting early estimates and falling into local optima. In this work, we leverage recent advances in variational Bayesian neural networks and combine these with traditional Deep Q-Networks (DQN) and Deep Deterministic Policy Gradient (DD…
Utrecht University（乌得勒支大学）；Georgia Institute of Technology（佐治亚理工学院）；Microsoft（微软）
Obtaining samples from the posterior distribution of inverse problems with expensive forward operators is challenging especially when the unknowns involve the strongly heterogeneous Earth. To meet these challenges, we propose a preconditioning scheme involving a conditional normalizing flow (NF) capable of sampling from a low-fidelity posterior distribution directly. This conditional NF is used to speed up the training of the high-fidelity objective involving minimization of the Kullback-Leibler divergence between the predicted and the desired high-fidelity posterior density for indirect measu…
