Spin-glasses are universal models that can capture complex behavior of many-body systems at the interface of statistical physics and computer science including discrete optimization, inference in graphical models, and automated reasoning. Computing the underlying structure and dynamics of such complex systems is extremely difficult due to the combinatorial explosion of their state space. Here, we develop deep generative continuous spin-glass distributions with normalizing flows to model correlations in generic discrete problems. We use a self-supervised learning paradigm by automatically generating the data from the spin-glass itself. We demonstrate that key physical and computational properties of the spin-glass phase can be successfully learned, including multi-modal steady-state distributions and topological structures among metastable states. Remarkably, we observe that the learning itself corresponds to a spin-glass phase transition within the layers of the trained normalizing flows. The inverse normalizing flows learns to perform reversible multi-scale coarse-graining operations which are very different from the typical irreversible renormalization group techniques.
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.
Heidelberg University（海德堡大学）；Bosch Center for Artificial Intelligence（博世人工智能中心）
The Information Bottleneck (IB) objective uses information theory to formulate a task-performance versus robustness trade-off. It has been successfully applied in the standard discriminative classification setting. We pose the question whether the IB can also be used to train generative likelihood models such as normalizing flows. Since normalizing flows use invertible network architectures (INNs), they are information-preserving by construction. This seems contradictory to the idea of a bottleneck. In this work, firstly, we develop the theory and methodology of IB-INNs, a class of conditional…
Deep generative models are a class of techniques that train deep neural networks to model the distribution of training samples. Research has fragmented into various interconnected approaches, each of which make trade-offs including run-time, diversity, and architectural restrictions. In particular, this compendium covers energy-based models, variational autoencoders, generative adversarial networks, autoregressive models, normalizing flows, in addition to numerous hybrid approaches. These techniques are compared and contrasted, explaining the premises behind each and how they are interrelated,…
University of Amsterdam（阿姆斯特丹大学）；Canadian Institute for Advanced Research (CIFAR)（加拿大高等研究院）
Generative flows are attractive because they admit exact likelihood optimization and efficient image synthesis. Recently, Kingma & Dhariwal (2018) demonstrated with Glow that generative flows are capable of generating high quality images. We generalize the 1 x 1 convolutions proposed in Glow to invertible d x d convolutions, which are more flexible since they operate on both channel and spatial axes. We propose two methods to produce invertible convolutions that have receptive fields identical to standard convolutions: Emerging convolutions are obtained by chaining specific autoregressive conv…
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…
Normalizing flows transform a latent distribution through an invertible neural network for a flexible and pleasingly simple approach to generative modelling, while preserving an exact likelihood. We propose FlowGMM, an end-to-end approach to generative semi supervised learning with normalizing flows, using a latent Gaussian mixture model. FlowGMM is distinct in its simplicity, unified treatment of labelled and unlabelled data with an exact likelihood, interpretability, and broad applicability beyond image data. We show promising results on a wide range of applications, including AG-News and Ya…
Generative models can have distinct mode of failures like mode dropping and low quality samples, which cannot be captured by a single scalar metric. To address this, recent works propose evaluating generative models using precision and recall, where precision measures quality of samples and recall measures the coverage of the target distribution. Although a variety of discrepancy measures between the target and estimated distribution are used to train generative models, it is unclear what precision-recall trade-offs are achieved by various choices of the discrepancy measures. In this paper, we…
Normalizing flows are a powerful class of generative models for continuous random variables, showing both strong model flexibility and the potential for non-autoregressive generation. These benefits are also desired when modeling discrete random variables such as text, but directly applying normalizing flows to discrete sequences poses significant additional challenges. We propose a VAE-based generative model which jointly learns a normalizing flow-based distribution in the latent space and a stochastic mapping to an observed discrete space. In this setting, we find that it is crucial for the …
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）
Greater direct electrification of end-use sectors with a higher share of renewables is one of the pillars to power a carbon-neutral society by 2050. However, in contrast to conventional power plants, renewable energy is subject to uncertainty raising challenges for their interaction with power systems. Scenario-based probabilistic forecasting models have become a vital tool to equip decision-makers. This paper presents to the power systems forecasting practitioners a recent deep learning technique, the normalizing flows, to produce accurate scenario-based probabilistic forecasts that are cruci…
Jet Propulsion Laboratory, California Institute of Technology（加州理工学院喷气推进实验室）
Firefighters suffer a variety of life-threatening risks, including line-of-duty deaths, injuries, and exposures to hazardous substances. Support for reducing these risks is important. We built a partially occluded object reconstruction method on augmented reality glasses for first responders. We used a deep learning based on conditional generative adversarial networks to train associations between the various images of flammable and hazardous objects and their partially occluded counterparts. Our system then reconstructed an image of a new flammable object. Finally, the reconstructed image was…
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…
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…
Entropy-Informed Weighting Channel Normalizing Flow for Deep Generative Models
School of Mathematics, South China University of Technology, Guangzhou, China（华南理工大学数学学院）；Shenzhen International Graduate School, Tsinghua University, Shenzhen, China（清华大学深圳国际研究生院）；School of Electronic and Information Engineering, South China University of Technology, Guangzhou, China（华南理工大学电子信息学院）；Department of Electrical Engineering, Columbia University, New York, NY USA（哥伦比亚大学电气工程系）
A popular approach to decrease the need for costly manual annotation of large data sets is weak supervision, which introduces problems of noisy labels, coverage and bias. Methods for overcoming these problems have either relied on discriminative models, trained with cost functions specific to weak supervision, and more recently, generative models, trying to model the output of the automatic annotation process. In this work, we explore a novel direction of generative modeling for weak supervision: Instead of modeling the output of the annotation process (the labeling function matches), we gener…
University of Toronto（多伦多大学）；Vector Institute（矢量研究院）；UC Berkeley（加州大学伯克利分校）；Google Research（谷歌研究院）
We introduce graph normalizing flows: a new, reversible graph neural network model for prediction and generation. On supervised tasks, graph normalizing flows perform similarly to message passing neural networks, but at a significantly reduced memory footprint, allowing them to scale to larger graphs. In the unsupervised case, we combine graph normalizing flows with a novel graph auto-encoder to create a generative model of graph structures. Our model is permutation-invariant, generating entire graphs with a single feed-forward pass, and achieves competitive results with the state-of-the art a…
Catastrophic forgetting (CF) happens whenever a neural network overwrites past knowledge while being trained on new tasks. Common techniques to handle CF include regularization of the weights (using, e.g., their importance on past tasks), and rehearsal strategies, where the network is constantly re-trained on past data. Generative models have also been applied for the latter, in order to have endless sources of data. In this paper, we propose a novel method that combines the strengths of regularization and generative-based rehearsal approaches. Our generative model consists of a normalizing fl…
In pursuit of explainability, we develop generative models for sequential data. The proposed models provide state-of-the-art classification results and robust performance for speech phone classification. We combine modern neural networks (normalizing flows) and traditional generative models (hidden Markov models - HMMs). Normalizing flow-based mixture models (NMMs) are used to model the conditional probability distribution given the hidden state in the HMMs. Model parameters are learned through judicious combinations of time-tested Bayesian learning methods and contemporary neural network lear…
Yandex（扬德克斯）；National Research University Higher School of Economics（国立研究型大学高等经济学院）
Normalizing flows are a powerful class of generative models demonstrating strong performance in several speech and vision problems. In contrast to other generative models, normalizing flows are latent variable models with tractable likelihoods and allow for stable training. However, they have to be carefully designed to represent invertible functions with efficient Jacobian determinant calculation. In practice, these requirements lead to overparameterized and sophisticated architectures that are inferior to alternative feed-forward models in terms of inference time and memory consumption. In t…
In this work, we propose a novel generative model for mapping inputs to structured, high-dimensional outputs using structured conditional normalizing flows and Gaussian process regression. The model is motivated by the need to characterize uncertainty in the input/output relationship when making inferences on new data. In particular, in the physical sciences, limited training data may not adequately characterize future observed data; it is critical that models adequately indicate uncertainty, particularly when they may be asked to extrapolate. In our proposed model, structured conditional norm…
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.
Normalizing flows have received a great deal of recent attention as they allow flexible generative modeling as well as easy likelihood computation. While a wide variety of flow models have been proposed, there is little formal understanding of the representation power of these models. In this work, we study some basic normalizing flows and rigorously establish bounds on their expressive power. Our results indicate that while these flows are highly expressive in one dimension, in higher dimensions their representation power may be limited, especially when the flows have moderate depth.
Given datasets from multiple domains, a key challenge is to efficiently exploit these data sources for modeling a target domain. Variants of this problem have been studied in many contexts, such as cross-domain translation and domain adaptation. We propose AlignFlow, a generative modeling framework that models each domain via a normalizing flow. The use of normalizing flows allows for a) flexibility in specifying learning objectives via adversarial training, maximum likelihood estimation, or a hybrid of the two methods; and b) learning and exact inference of a shared representation in the late…
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…
SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flows
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（HISKP））；Oldendorff Carriers GmbH & Co. KG（奥尔登多夫航运有限两合公司）
Mila（米拉）；Université de Montréal（蒙特利尔大学）；McGill University（麦吉尔大学）；Deep Render（深度渲染公司）；Polytechnique Montréal（蒙特利尔综合理工学院）
Recent work has shown that Neural Ordinary Differential Equations (ODEs) can serve as generative models of images using the perspective of Continuous Normalizing Flows (CNFs). Such models offer exact likelihood calculation, and invertible generation/density estimation. In this work we introduce a Multi-Resolution variant of such models (MRCNF), by characterizing the conditional distribution over the additional information required to generate a fine image that is consistent with the coarse image. We introduce a transformation between resolutions that allows for no change in the log likelihood.…
We present a generative model that is defined on finite sets of exchangeable, potentially high dimensional, data. As the architecture is an extension of RealNVPs, it inherits all its favorable properties, such as being invertible and allowing for exact log-likelihood evaluation. We show that this architecture is able to learn finite non-i.i.d. set data distributions, learn statistical dependencies between entities of the set and is able to train and sample with variable set sizes in a computationally efficient manner. Experiments on 3D point clouds show state-of-the art likelihoods.
National Yang Ming Chiao Tung University（国立阳明交通大学）；CNIT（意大利国家电信联盟）；University of Brescia（布雷西亚大学）
This paper presents an end-to-end learning-based video compression system, termed CANF-VC, based on conditional augmented normalizing flows (CANF). Most learned video compression systems adopt the same hybrid-based coding architecture as the traditional codecs. Recent research on conditional coding has shown the sub-optimality of the hybrid-based coding and opens up opportunities for deep generative models to take a key role in creating new coding frameworks. CANF-VC represents a new attempt that leverages the conditional ANF to learn a video generative model for conditional inter-frame coding…
York University（约克大学）；Borealis AI（北极光人工智能）；Ryerson University（瑞尔森大学）；Samsung AI Centre Toronto（三星人工智能中心多伦多分部）；Vector Institute（向量研究所）
Normalizing flows are a class of probabilistic generative models which allow for both fast density computation and efficient sampling and are effective at modelling complex distributions like images. A drawback among current methods is their significant training cost, sometimes requiring months of GPU training time to achieve state-of-the-art results. This paper introduces Wavelet Flow, a multi-scale, normalizing flow architecture based on wavelets. A Wavelet Flow has an explicit representation of signal scale that inherently includes models of lower resolution signals and conditional generati…
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）；University of Oxford（牛津大学）
This paper introduces a generative model equivariant to Euclidean symmetries: E(n) Equivariant Normalizing Flows (E-NFs). To construct E-NFs, we take the discriminative E(n) graph neural networks and integrate them as a differential equation to obtain an invertible equivariant function: a continuous-time normalizing flow. We demonstrate that E-NFs considerably outperform baselines and existing methods from the literature on particle systems such as DW4 and LJ13, and on molecules from QM9 in terms of log-likelihood. To the best of our knowledge, this is the first flow that jointly generates mol…
CSEM（瑞士电子与微技术中心）；University of Maryland（马里兰大学）；ETHZ（苏黎世联邦理工学院）
Learning probabilistic models that can estimate the density of a given set of samples, and generate samples from that density, is one of the fundamental challenges in unsupervised machine learning. We introduce a new generative model based on denoising density estimators (DDEs), which are scalar functions parameterized by neural networks, that are efficiently trained to represent kernel density estimators of the data. Leveraging DDEs, our main contribution is a novel technique to obtain generative models by minimizing the KL-divergence directly. We prove that our algorithm for obtaining genera…
Weizmann Institute of Science（魏茨曼科学研究所）；University College London（伦敦大学学院）；Meta AI Research（Meta人工智能研究院）
Continuous Normalizing Flows (CNFs) are a class of generative models that transform a prior distribution to a model distribution by solving an ordinary differential equation (ODE). We propose to train CNFs on manifolds by minimizing probability path divergence (PPD), a novel family of divergences between the probability density path generated by the CNF and a target probability density path. PPD is formulated using a logarithmic mass conservation formula which is a linear first order partial differential equation relating the log target probabilities and the CNF's defining vector field. PPD ha…
Hasso Plattner Institute for Digital Engineering（哈索·普拉特纳数字工程研究所）；University of Potsdam（波茨坦大学）；University of Kaiserslautern-Landau（凯泽斯劳滕-兰道大学）；Hasso Plattner Institute for Digital Health at the Icahn School of Medicine at Mount Sinai（西奈山伊坎医学院哈索·普拉特纳数字健康研究所）
Despite recent advances in generative modeling for text-to-speech synthesis, these models do not yet have the same fine-grained adjustability of pitch-conditioned deterministic models such as FastPitch and FastSpeech2. Pitch information is not only low-dimensional, but also discontinuous, making it particularly difficult to model in a generative setting. Our work explores several techniques for handling the aforementioned issues in the context of Normalizing Flow models. We also find this problem to be very well suited for Neural Spline flows, which is a highly expressive alternative to the mo…
Detecting out-of-distribution (OOD) data is crucial for robust machine learning systems. Normalizing flows are flexible deep generative models that often surprisingly fail to distinguish between in- and out-of-distribution data: a flow trained on pictures of clothing assigns higher likelihood to handwritten digits. We investigate why normalizing flows perform poorly for OOD detection. We demonstrate that flows learn local pixel correlations and generic image-to-latent-space transformations which are not specific to the target image dataset. We show that by modifying the architecture of flow co…
In recent years, various flow-based generative models have been proposed to generate high-fidelity waveforms in real-time. However, these models require either a well-trained teacher network or a number of flow steps making them memory-inefficient. In this paper, we propose a novel generative model called WaveNODE which exploits a continuous normalizing flow for speech synthesis. Unlike the conventional models, WaveNODE places no constraint on the function used for flow operation, thus allowing the usage of more flexible and complex functions. Moreover, WaveNODE can be optimized to maximize th…
Université Paris Cité（巴黎西岱大学）；INRIA（法国国家信息与自动化研究所）；Inserm（法国国家健康与医学研究院）；Centre de Recherche des Cordeliers（科德利耶研究中心）
This paper introduces a new latent variable generative model able to handle high dimensional longitudinal data and relying on variational inference. The time dependency between the observations of an input sequence is modelled using normalizing flows over the associated latent variables. The proposed method can be used to generate either fully synthetic longitudinal sequences or trajectories that are conditioned on several data in a sequence and demonstrates good robustness properties to missing data. We test the model on 6 datasets of different complexity and show that it can achieve better l…
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 op…
The two key characteristics of a normalizing flow is that it is invertible (in particular, dimension preserving) and that it monitors the amount by which it changes the likelihood of data points as samples are propagated along the network. Recently, multiple generalizations of normalizing flows have been introduced that relax these two conditions. On the other hand, neural networks only perform a forward pass on the input, there is neither a notion of an inverse of a neural network nor is there one of its likelihood contribution. In this paper we argue that certain neural network architectures…
TrajFlow: A Generative Framework for Occupancy Density Estimation Using Normalizing Flows
Department of Computer Science University of Minnesota（计算机科学系明尼苏达大学）；Department of Civil Environmental and Geo- Engineering University of Minnesota（土木环境与地球工程系明尼苏达大学）
University of Adelaide（阿德莱德大学）；School of Computer Science（计算机科学学院）
In this paper, we propose a score-based normalizing flow method called DAG-NF to learn dependencies of input observation data. Inspired by Grad-CAM in computer vision, we use jacobian matrix of output on input as causal relationships and this method can be generalized to any neural networks especially for flow-based generative neural networks such as Masked Autoregressive Flow(MAF) and Continuous Normalizing Flow(CNF) which compute the log likelihood loss and divergence of distribution of input data and target distribution. This method extends NOTEARS which enforces a important acylicity const…
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 …
Generative modeling seeks to uncover the underlying factors that give rise to observed data that can often be modeled as the natural symmetries that manifest themselves through invariances and equivariances to certain transformation laws. However, current approaches to representing these symmetries are couched in the formalism of continuous normalizing flows that require the construction of equivariant vector fields -- inhibiting their simple application to conventional higher dimensional generative modelling domains like natural images. In this paper, we focus on building equivariant normaliz…
Based on the manifold hypothesis, real-world data often lie on a low-dimensional manifold, while normalizing flows as a likelihood-based generative model are incapable of finding this manifold due to their structural constraints. So, one interesting question arises: $\textit{"Can we find sub-manifold(s) of data in normalizing flows and estimate the density of the data on the sub-manifold(s)?"}$. In this paper, we introduce two approaches, namely per-pixel penalized log-likelihood and hierarchical training, to answer the mentioned question. We propose a single-step method for joint manifold lea…
An approach to utilize recent advances in deep generative models for anomaly detection in a granular (continuous) sense on a real-world image dataset with quality issues is detailed using recent normalizing flow models, with implications in many other applications/domains/data types. The approach is completely unsupervised (no annotations available) but qualitatively shown to provide accurate semantic labeling for images via heatmaps of the scaled log-likelihood overlaid on the images. When sorted based on the median values per image, clear trends in quality are observed. Furthermore, downstre…
Normalizing flows, diffusion normalizing flows and variational autoencoders are powerful generative models. This chapter provides a unified framework to handle these approaches via Markov chains. We consider stochastic normalizing flows as a pair of Markov chains fulfilling some properties and show how many state-of-the-art models for data generation fit into this framework. Indeed numerical simulations show that including stochastic layers improves the expressivity of the network and allows for generating multimodal distributions from unimodal ones. The Markov chains point of view enables us …
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…
