Normalizing flows are constructed from a base distribution with a known density and a diffeomorphism with a tractable Jacobian. The base density of a normalizing flow can be parameterised by a different normalizing flow, thus allowing maps to be found between arbitrary distributions. We demonstrate and explore the utility of this approach and show it is particularly interesting in the case of conditional normalizing flows and for introducing optimal transport constraints on maps that are constructed using normalizing flows.
University of Toulouse（图卢兹大学）；Institut Universitaire de France（法国大学研究院）；Univ. Lille（里尔大学）；CNRS（法国国家科学研究中心）；Centrale Lille（里尔中央理工学院）
Despite their advantages, normalizing flows generally suffer from several shortcomings including their tendency to generate unrealistic data (e.g., images) and their failing to detect out-of-distribution data. One reason for these deficiencies lies in the training strategy which traditionally exploits a maximum likelihood principle only. This paper proposes a new training paradigm based on a hybrid objective function combining the maximum likelihood principle (MLE) and a sliced-Wasserstein distance. Results obtained on synthetic toy examples and real image data sets show better generative abil…
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…
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…
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.…
Forschungszentrum Jülich GmbH（于利希研究中心）；RWTH Aachen University（亚琛工业大学）；JARA-ENERGY（于利希-亚琛研究联盟能源组）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
Neural networks-based learning of the distribution of non-dispatchable renewable electricity generation from sources such as photovoltaics (PV) and wind as well as load demands has recently gained attention. Normalizing flow density models are particularly well suited for this task due to the training through direct log-likelihood maximization. However, research from the field of image generation has shown that standard normalizing flows can only learn smeared-out versions of manifold distributions. Previous works on normalizing flow-based scenario generation do not address this issue, and the…
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…
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…
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…
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.
Normalizing flows learn a diffeomorphic mapping between the target and base distribution, while the Jacobian determinant of that mapping forms another real-valued function. In this paper, we show that the Jacobian determinant mapping is unique for the given distributions, hence the likelihood objective of flows has a unique global optimum. In particular, the likelihood for a class of flows is explicitly expressed by the eigenvalues of the auto-correlation matrix of individual data point, and independent of the parameterization of neural network, which provides a theoretical optimal value of li…
University of California at Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
Our universe is homogeneous and isotropic, and its perturbations obey translation and rotation symmetry. In this work we develop Translation and Rotation Equivariant Normalizing Flow (TRENF), a generative Normalizing Flow (NF) model which explicitly incorporates these symmetries, defining the data likelihood via a sequence of Fourier space-based convolutions and pixel-wise nonlinear transforms. TRENF gives direct access to the high dimensional data likelihood p(x|y) as a function of the labels y, such as cosmological parameters. In contrast to traditional analyses based on summary statistics, …
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 …
University of Waterloo（滑铁卢大学）；Vector Institute（矢量研究所）；University of Edinburgh（爱丁堡大学）
Density deconvolution is the task of estimating a probability density function given only noise-corrupted samples. We can fit a Gaussian mixture model to the underlying density by maximum likelihood if the noise is normally distributed, but would like to exploit the superior density estimation performance of normalizing flows and allow for arbitrary noise distributions. Since both adjustments lead to an intractable likelihood, we resort to amortized variational inference. We demonstrate some problems involved in this approach, however, experiments on real data demonstrate that flows can alread…
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…
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…
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）
Mapping between discrete and continuous distributions is a difficult task and many have had to resort to heuristical approaches. We propose a tessellation-based approach that directly learns quantization boundaries in a continuous space, complete with exact likelihood evaluations. This is done through constructing normalizing flows on convex polytopes parameterized using a simple homeomorphism with an efficient log determinant Jacobian. We explore this approach in two application settings, mapping from discrete to continuous and vice versa. Firstly, a Voronoi dequantization allows automaticall…
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 …
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…
fortiss GmbH, Research Institute of the Free State of Bavaria（巴伐利亚自由州研究机构fortiss有限公司）；Technical University of Munich（慕尼黑工业大学）
The future motion of traffic participants is inherently uncertain. To plan safely, therefore, an autonomous agent must take into account multiple possible trajectory outcomes and prioritize them. Recently, this problem has been addressed with generative neural networks. However, most generative models either do not learn the true underlying trajectory distribution reliably, or do not allow predictions to be associated with likelihoods. In our work, we model motion prediction directly as a density estimation problem with a normalizing flow between a noise distribution and the future motion dist…
Federal University of Ceará（塞阿拉联邦大学）；National Department of Public Security（国家公共安全部）
The task of detecting anomalous data patterns is as important in practical applications as challenging. In the context of spatial data, recognition of unexpected trajectories brings additional difficulties, such as high dimensionality and varying pattern lengths. We aim to tackle such a problem from a probability density estimation point of view, since it provides an unsupervised procedure to identify out of distribution samples. More specifically, we pursue an approach based on normalizing flows, a recent framework that enables complex density estimation from data with neural networks. Our pr…
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.
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.
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…
The detection of manufacturing errors is crucial in fabrication processes to ensure product quality and safety standards. Since many defects occur very rarely and their characteristics are mostly unknown a priori, their detection is still an open research question. To this end, we propose DifferNet: It leverages the descriptiveness of features extracted by convolutional neural networks to estimate their density using normalizing flows. Normalizing flows are well-suited to deal with low dimensional data distributions. However, they struggle with the high dimensionality of images. Therefore, we …
University of California at Berkeley（加州大学伯克利分校）；International Computer Science Institute（国际计算机科学研究所）；University of Queensland（昆士兰大学）
We introduce stochastic normalizing flows, an extension of continuous normalizing flows for maximum likelihood estimation and variational inference (VI) using stochastic differential equations (SDEs). Using the theory of rough paths, the underlying Brownian motion is treated as a latent variable and approximated, enabling efficient training of neural SDEs as random neural ordinary differential equations. These SDEs can be used for constructing efficient Markov chains to sample from the underlying distribution of a given dataset. Furthermore, by considering families of targeted SDEs with prescr…
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…
Coupling-based normalizing flows (e.g. RealNVP) are a popular family of normalizing flow architectures that work surprisingly well in practice. This calls for theoretical understanding. Existing work shows that such flows weakly converge to arbitrary data distributions. However, they make no statement about the stricter convergence criterion used in practice, the maximum likelihood loss. For the first time, we make a quantitative statement about this kind of convergence: We prove that all coupling-based normalizing flows perform whitening of the data distribution (i.e. diagonalize the covarian…
The latent space of normalizing flows must be of the same dimensionality as their output space. This constraint presents a problem if we want to learn low-dimensional, semantically meaningful representations. Recent work has provided compact representations by fitting flows constrained to manifolds, but hasn't defined a density off that manifold. In this work we consider flows with full support in data space, but with ordered latent variables. Like in PCA, the leading latent dimensions define a sequence of manifolds that lie close to the data. We note a trade-off between the flow likelihood an…
Normalizing flows are among the most popular paradigms in generative modeling, especially for images, primarily because we can efficiently evaluate the likelihood of a data point. This is desirable both for evaluating the fit of a model, and for ease of training, as maximizing the likelihood can be done by gradient descent. However, training normalizing flows comes with difficulties as well: models which produce good samples typically need to be extremely deep -- which comes with accompanying vanishing/exploding gradient problems. A very related problem is that they are often poorly conditione…
Anomaly detection in time-series via inductive biases in the latent space of conditional normalizing flows
Department of Computer Science Norwegian University of Science and Technology（挪威科学技术大学计算机科学系）；BCAM – Basque Center for Applied Mathematics University of Coimbra, CISUC/LASI, DEI（巴斯克应用数学中心，科英布拉大学，CISUC/LASI，DEI）；IKERBASQUE — Basque Foundation for Science（巴斯克科学基金会）
We test the robustness of a maximum-likelihood (ML) based classifier where sequential data as observation is corrupted by noise. The hypothesis is that a generative model, that combines the state transitions of a hidden Markov model (HMM) and the neural network based probability distributions for the hidden states of the HMM, can provide a robust classification performance. The combined model is called normalizing-flow mixture model based HMM (NMM-HMM). It can be trained using a combination of expectation-maximization (EM) and backpropagation. We verify the improved robustness of NMM-HMM class…
There exists a high variability in mobility data volumes across different regions, which deteriorates the performance of spatial recommender systems that rely on region-specific data. In this paper, we propose a novel transfer learning framework called REFORMD, for continuous-time location prediction for regions with sparse checkin data. Specifically, we model user-specific checkin-sequences in a region using a marked temporal point process (MTPP) with normalizing flows to learn the inter-checkin time and geo-distributions. Later, we transfer the model parameters of spatial and temporal flows …
Normalizing flows are a class of flexible deep generative models that offer easy likelihood computation. Despite their empirical success, there is little theoretical understanding of their expressiveness. In this work, we study residual flows, a class of normalizing flows composed of Lipschitz residual blocks. We prove residual flows are universal approximators in maximum mean discrepancy. We provide upper bounds on the number of residual blocks to achieve approximation under different assumptions.
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…
