Despite their popularity, to date, the application of normalizing flows on categorical data stays limited. The current practice of using dequantization to map discrete data to a continuous space is inapplicable as categorical data has no intrinsic order. Instead, categorical data have complex and latent relations that must be inferred, like the synonymy between words. In this paper, we investigate \emph{Categorical Normalizing Flows}, that is normalizing flows for categorical data. By casting the encoding of categorical data in continuous space as a variational inference problem, we jointly optimize the continuous representation and the model likelihood. Using a factorized decoder, we introduce an inductive bias to model any interactions in the normalizing flow. As a consequence, we do not only simplify the optimization compared to having a joint decoder, but also make it possible to scale up to a large number of categories that is currently impossible with discrete normalizing flows. Based on Categorical Normalizing Flows, we propose GraphCNF a permutation-invariant generative model on graphs. GraphCNF implements a three step approach modeling the nodes, edges and adjacency matrix stepwise to increase efficiency. On molecule generation, GraphCNF outperforms both one-shot and autoregressive flow-based state-of-the-art.
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…
Normalizing flows have emerged as an important family of deep neural networks for modelling complex probability distributions. In this note, we revisit their coupling and autoregressive transformation layers as probabilistic graphical models and show that they reduce to Bayesian networks with a pre-defined topology and a learnable density at each node. From this new perspective, we provide three results. First, we show that stacking multiple transformations in a normalizing flow relaxes independence assumptions and entangles the model distribution. Second, we show that a fundamental leap of ca…
Normalizing flows model complex probability distributions by combining a base distribution with a series of bijective neural networks. State-of-the-art architectures rely on coupling and autoregressive transformations to lift up invertible functions from scalars to vectors. In this work, we revisit these transformations as probabilistic graphical models, showing they reduce to Bayesian networks with a pre-defined topology and a learnable density at each node. From this new perspective, we propose the graphical normalizing flow, a new invertible transformation with either a prescribed or a lear…
Technical University of Denmark（丹麦技术大学）；UvA-Bosch Delta Lab（阿姆斯特丹大学-博世三角洲实验室）；University of Amsterdam（阿姆斯特丹大学）
Normalizing flows and variational autoencoders are powerful generative models that can represent complicated density functions. However, they both impose constraints on the models: Normalizing flows use bijective transformations to model densities whereas VAEs learn stochastic transformations that are non-invertible and thus typically do not provide tractable estimates of the marginal likelihood. In this paper, we introduce SurVAE Flows: A modular framework of composable transformations that encompasses VAEs and normalizing flows. SurVAE Flows bridge the gap between normalizing flows and VAEs …
CONTRA: Conformal Prediction Region via Normalizing Flow Transformation
Department of Statistics and Actuarial Science（统计与精算科学系）；The University of Iowa（爱荷华大学）；Department of Applied Mathematics（应用数学系）；The Hong Kong Polytechnic University（香港理工大学）
Real-world data with underlying structure, such as pictures of faces, are hypothesized to lie on a low-dimensional manifold. This manifold hypothesis has motivated state-of-the-art generative algorithms that learn low-dimensional data representations. Unfortunately, a popular generative model, normalizing flows, cannot take advantage of this. Normalizing flows are based on successive variable transformations that are, by design, incapable of learning lower-dimensional representations. In this paper we introduce noisy injective flows (NIF), a generalization of normalizing flows that can go acro…
Fractal Flow: Hierarchical and Interpretable Normalizing Flow via Topic Modeling and Recursive Strategy
School of Mathematics Harbin Institute of Technology（哈尔滨工业大学数学学院）；School of Mathematics and Institute for Artificial Intelligence Harbin Instiute of Technology（哈尔滨工业大学数学学院和人工智能研究院）；School of Earth and Space Sciences and Institute for Artificial Intelligence Peking University（北京大学地球和空间科学学院和人工智能研究院）
Normalizing flows provide a general mechanism for defining expressive probability distributions, only requiring the specification of a (usually simple) base distribution and a series of bijective transformations. There has been much recent work on normalizing flows, ranging from improving their expressive power to expanding their application. We believe the field has now matured and is in need of a unified perspective. In this review, we attempt to provide such a perspective by describing flows through the lens of probabilistic modeling and inference. We place special emphasis on the fundament…
Normalizing flows are a class of deep generative models that provide a promising route to sample lattice field theories more efficiently than conventional Monte Carlo simulations. In this work we show that the theoretical framework of stochastic normalizing flows, in which neural-network layers are combined with Monte Carlo updates, is the same that underlies out-of-equilibrium simulations based on Jarzynski's equality, which have been recently deployed to compute free-energy differences in lattice gauge theories. We lay out a strategy to optimize the efficiency of this extended class of gener…
University of Amsterdam（阿姆斯特丹大学）；Technical University of Denmark（丹麦技术大学）
Generative flows and diffusion models have been predominantly trained on ordinal data, for example natural images. This paper introduces two extensions of flows and diffusion for categorical data such as language or image segmentation: Argmax Flows and Multinomial Diffusion. Argmax Flows are defined by a composition of a continuous distribution (such as a normalizing flow), and an argmax function. To optimize this model, we learn a probabilistic inverse for the argmax that lifts the categorical data to a continuous space. Multinomial Diffusion gradually adds categorical noise in a diffusion pr…
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…
Columbia University（哥伦比亚大学）；Layer6 AI（Layer6人工智能公司）；New York University（纽约大学）
Probability distributions supported on the simplex enjoy a wide range of applications across statistics and machine learning. Recently, a novel family of such distributions has been discovered: the continuous categorical. This family enjoys remarkable mathematical simplicity; its density function resembles that of the Dirichlet distribution, but with a normalizing constant that can be written in closed form using elementary functions only. In spite of this mathematical simplicity, our understanding of the normalizing constant remains far from complete. In this work, we characterize the numeric…
Department of Physics（物理系）；The Hong Kong University of Science and Technology（香港科技大学）
Wavelet transformation stands as a cornerstone in modern data analysis and signal processing. Its mathematical essence is an invertible transformation that discerns slow patterns from fast ones in the frequency domain. Such an invertible transformation can be learned by a designed normalizing flow model. With a generalized lifting scheme as coupling layers, a factor-out layer resembling the downsampling, and parameter sharing at different levels of the model, one can train the normalizing flow to filter high-frequency elements at different levels, thus extending traditional linear wavelet tran…
Normalizing flows define a probability distribution by an explicit invertible transformation $\boldsymbol{\mathbf{z}}=f(\boldsymbol{\mathbf{x}})$. In this work, we present implicit normalizing flows (ImpFlows), which generalize normalizing flows by allowing the mapping to be implicitly defined by the roots of an equation $F(\boldsymbol{\mathbf{z}}, \boldsymbol{\mathbf{x}})= \boldsymbol{\mathbf{0}}$. ImpFlows build on residual flows (ResFlows) with a proper balance between expressiveness and tractability. Through theoretical analysis, we show that the function space of ImpFlow is strictly riche…
Modeling sets is an important problem in machine learning since this type of data can be found in many domains. A promising approach defines a family of permutation invariant densities with continuous normalizing flows. This allows us to maximize the likelihood directly and sample new realizations with ease. In this work, we demonstrate how calculating the trace, a crucial step in this method, raises issues that occur both during training and inference, limiting its practicality. We propose an alternative way of defining permutation equivariant transformations that give closed form trace. This…
McGill University（麦吉尔大学）；Mila（米拉研究所）；University of Toronto（多伦多大学）；Vector Institute（向量研究所）
The choice of approximate posterior distributions plays a central role in stochastic variational inference (SVI). One effective solution is the use of normalizing flows \cut{defined on Euclidean spaces} to construct flexible posterior distributions. However, one key limitation of existing normalizing flows is that they are restricted to the Euclidean space and are ill-equipped to model data with an underlying hierarchical structure. To address this fundamental limitation, we present the first extension of normalizing flows to hyperbolic spaces. We first elevate normalizing flows to hyperbolic …
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…
Normalizing flows are deep generative models that allow efficient likelihood calculation and sampling. The core requirement for this advantage is that they are constructed using functions that can be efficiently inverted and for which the determinant of the function's Jacobian can be efficiently computed. Researchers have introduced various such flow operations, but few of these allow rich interactions among variables without incurring significant computational costs. In this paper, we introduce Woodbury transformations, which achieve efficient invertibility via the Woodbury matrix identity an…
Normalizing flows (NFs) have become a prominent method for deep generative models that allow for an analytic probability density estimation and efficient synthesis. However, a flow-based network is considered to be inefficient in parameter complexity because of reduced expressiveness of bijective mapping, which renders the models unfeasibly expensive in terms of parameters. We present an alternative parameterization scheme called NanoFlow, which uses a single neural density estimator to model multiple transformation stages. Hence, we propose an efficient parameter decomposition method and the …
Normalizing flows attempt to model an arbitrary probability distribution through a set of invertible mappings. These transformations are required to achieve a tractable Jacobian determinant that can be used in high-dimensional scenarios. The first normalizing flow designs used coupling layer mappings built upon affine transformations. The significant advantage of such models is their easy-to-compute inverse. Nevertheless, making use of affine transformations may limit the expressiveness of such models. Recently, invertible piecewise polynomial functions as a replacement for affine transformati…
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…
University of Amsterdam（阿姆斯特丹大学）；University of Oxford（牛津大学）
Variational inference relies on flexible approximate posterior distributions. Normalizing flows provide a general recipe to construct flexible variational posteriors. We introduce Sylvester normalizing flows, which can be seen as a generalization of planar flows. Sylvester normalizing flows remove the well-known single-unit bottleneck from planar flows, making a single transformation much more flexible. We compare the performance of Sylvester normalizing flows against planar flows and inverse autoregressive flows and demonstrate that they compare favorably on several datasets.
Flow-based deep generative models learn data distributions by transforming a simple base distribution into a complex distribution via a set of invertible transformations. Due to the invertibility, such models can score unseen data samples by computing their exact likelihood under the learned distribution. This makes flow-based models a perfect tool for novelty detection, an anomaly detection technique where unseen data samples are classified as normal or abnormal by scoring them against a learned model of normal data. We show that normalizing flows can be used as novelty detectors in time seri…
Freie Universität Berlin（柏林自由大学）；Rice University（莱斯大学）
Normalizing flows are a promising tool for modeling probability distributions in physical systems. While state-of-the-art flows accurately approximate distributions and energies, applications in physics additionally require smooth energies to compute forces and higher-order derivatives. Furthermore, such densities are often defined on non-trivial topologies. A recent example are Boltzmann Generators for generating 3D-structures of peptides and small proteins. These generative models leverage the space of internal coordinates (dihedrals, angles, and bonds), which is a product of hypertori and c…
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 …
This paper studies the cooperative learning of two generative flow models, in which the two models are iteratively updated based on the jointly synthesized examples. The first flow model is a normalizing flow that transforms an initial simple density to a target density by applying a sequence of invertible transformations. The second flow model is a Langevin flow that runs finite steps of gradient-based MCMC toward an energy-based model. We start from proposing a generative framework that trains an energy-based model with a normalizing flow as an amortized sampler to initialize the MCMC chains…
University of Technology Sydney（悉尼科技大学）；Curtin University（科廷大学）
Counterfactual explanation is a form of interpretable machine learning that generates perturbations on a sample to achieve the desired outcome. The generated samples can act as instructions to guide end users on how to observe the desired results by altering samples. Although state-of-the-art counterfactual explanation methods are proposed to use variational autoencoder (VAE) to achieve promising improvements, they suffer from two major limitations: 1) the counterfactuals generation is prohibitively slow, which prevents algorithms from being deployed in interactive environments; 2) the counter…
OpenAI；University of Amsterdam（阿姆斯特丹大学）；University of California Irvine（加利福尼亚大学尔湾分校）；Canadian Institute for Advanced Research（加拿大高等研究院）
The framework of normalizing flows provides a general strategy for flexible variational inference of posteriors over latent variables. We propose a new type of normalizing flow, inverse autoregressive flow (IAF), that, in contrast to earlier published flows, scales well to high-dimensional latent spaces. The proposed flow consists of a chain of invertible transformations, where each transformation is based on an autoregressive neural network. In experiments, we show that IAF significantly improves upon diagonal Gaussian approximate posteriors. In addition, we demonstrate that a novel type of v…
Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院）；Instituto de Telecomunicações（电信研究所）
In the past few years, deep generative models, such as generative adversarial networks \autocite{GAN}, variational autoencoders \autocite{vaepaper}, and their variants, have seen wide adoption for the task of modelling complex data distributions. In spite of the outstanding sample quality achieved by those early methods, they model the target distributions \emph{implicitly}, in the sense that the probability density functions induced by them are not explicitly accessible. This fact renders those methods unfit for tasks that require, for example, scoring new instances of data with the learned d…
Data augmentation is a widely adopted technique for avoiding overfitting when training deep neural networks. However, this approach requires domain-specific knowledge and is often limited to a fixed set of hard-coded transformations. Recently, several works proposed to use generative models for generating semantically meaningful perturbations to train a classifier. However, because accurate encoding and decoding are critical, these methods, which use architectures that approximate the latent-variable inference, remained limited to pilot studies on small datasets. Exploiting the exactly reversi…
Korea University（高丽大学）；Sungkyunkwan University（成均馆大学）
Normalizing flows model probability distributions by learning invertible transformations that transfer a simple distribution into complex distributions. Since the architecture of ResNet-based normalizing flows is more flexible than that of coupling-based models, ResNet-based normalizing flows have been widely studied in recent years. Despite their architectural flexibility, it is well-known that the current ResNet-based models suffer from constrained Lipschitz constants. In this paper, we propose the monotone formulation to overcome the issue of the Lipschitz constants using monotone operators…
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…
Conditional Normalizing Flows for Forward and Backward Joint State and Parameter Estimation
Department of Computer Science and Engineering University of Notre Dame（计算机科学与工程系诺特达姆大学）；Department of Pediatrics Stanford University（儿科系斯坦福大学）；Department of Applied and Computational Mathematics and Statistics University of Notre Dame（应用与计算数学与统计系诺特达姆大学）
Variational auto-encoders (VAE) are scalable and powerful generative models. However, the choice of the variational posterior determines tractability and flexibility of the VAE. Commonly, latent variables are modeled using the normal distribution with a diagonal covariance matrix. This results in computational efficiency but typically it is not flexible enough to match the true posterior distribution. One fashion of enriching the variational posterior distribution is application of normalizing flows, i.e., a series of invertible transformations to latent variables with a simple posterior. In t…
Bayesian posterior inference is prevalent in various machine learning problems. Variational inference provides one way to approximate the posterior distribution, however its expressive power is limited and so is the accuracy of resulting approximation. Recently, there has a trend of using neural networks to approximate the variational posterior distribution due to the flexibility of neural network architecture. One way to construct flexible variational distribution is to warp a simple density into a complex by normalizing flows, where the resulting density can be analytically evaluated. Howeve…
Duke University（杜克大学）；Carnegie Mellon University（卡内基梅隆大学）
Normalizing flows are a widely used class of latent-variable generative models with a tractable likelihood. Affine-coupling (Dinh et al, 2014-16) models are a particularly common type of normalizing flows, for which the Jacobian of the latent-to-observable-variable transformation is triangular, allowing the likelihood to be computed in linear time. Despite the widespread usage of affine couplings, the special structure of the architecture makes understanding their representational power challenging. The question of universal approximation was only recently resolved by three parallel papers (Hu…
Emory University（埃默里大学）；University of Kentucky（肯塔基大学）
Nonlinear monotone transformations are used extensively in normalizing flows to construct invertible triangular mappings from simple distributions to complex ones. In existing literature, monotonicity is usually enforced by restricting function classes or model parameters and the inverse transformation is often approximated by root-finding algorithms as a closed-form inverse is unavailable. In this paper, we introduce a new integral-based approach termed "Atomic Unrestricted Time Machine (AUTM)", equipped with unrestricted integrands and easy-to-compute explicit inverse. AUTM offers a versatil…
Normalizing flows map an independent set of latent variables to their samples using a bijective transformation. Despite the exact correspondence between samples and latent variables, their high level relationship is not well understood. In this paper we characterize the geometric structure of flows using principal manifolds and understand the relationship between latent variables and samples using contours. We introduce a novel class of normalizing flows, called principal manifold flows (PF), whose contours are its principal manifolds, and a variant for injective flows (iPF) that is more effic…
Argonne National Laboratory（阿贡国家实验室）；Brookhaven National Laboratory（布鲁克海文国家实验室）；University of Connecticut（康涅狄格大学）
We propose using Normalizing Flows as a trainable kernel within the molecular dynamics update of Hamiltonian Monte Carlo (HMC). By learning (invertible) transformations that simplify our dynamics, we can outperform traditional methods at generating independent configurations. We show that, using a carefully constructed network architecture, our approach can be easily scaled to large lattice volumes with minimal retraining effort. The source code for our implementation is publicly available online at https://github.com/nftqcd/fthmc.
How to generate conditional synthetic data for a domain without utilizing information about its labels/attributes? Our work presents a solution to the above question. We propose a transfer learning-based framework utilizing normalizing flows, coupled with both maximum-likelihood and adversarial training. We model a source domain (labels available) and a target domain (labels unavailable) with individual normalizing flows, and perform domain alignment to a common latent space using adversarial discriminators. Due to the invertible property of flow models, the mapping has exact cycle consistency…
Tencent（腾讯）；National University of Singapore（新加坡国立大学）；MIT（麻省理工学院）
Deep Gaussian processes (DGPs), a hierarchical composition of GP models, have successfully boosted the expressive power of their single-layer counterpart. However, it is impossible to perform exact inference in DGPs, which has motivated the recent development of variational inference-based methods. Unfortunately, either these methods yield a biased posterior belief or it is difficult to evaluate their convergence. This paper introduces a new approach for specifying flexible, arbitrarily complex, and scalable approximate posterior distributions. The posterior distribution is constructed through…
OnePlanet Research Center（OnePlanet研究中心）；imec-the Netherlands（荷兰微电子研究中心（imec荷兰分部））；Google Research（谷歌研究院）；Donders Institute for Brain, Cognition and Behaviour（唐德斯大脑、认知与行为研究所）；Radboud University（拉德堡德大学）
Normalizing flows have shown great success as general-purpose density estimators. However, many real world applications require the use of domain-specific knowledge, which normalizing flows cannot readily incorporate. We propose embedded-model flows (EMF), which alternate general-purpose transformations with structured layers that embed domain-specific inductive biases. These layers are automatically constructed by converting user-specified differentiable probabilistic models into equivalent bijective transformations. We also introduce gated structured layers, which allow bypassing the parts o…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）
Deep learning models suffer from a phenomenon called adversarial attacks: we can apply minor changes to the model input to fool a classifier for a particular example. The literature mostly considers adversarial attacks on models with images and other structured inputs. However, the adversarial attacks for categorical sequences can also be harmful. Successful attacks for inputs in the form of categorical sequences should address the following challenges: (1) non-differentiability of the target function, (2) constraints on transformations of initial sequences, and (3) diversity of possible probl…
