An invertible function is bi-Lipschitz if both the function and its inverse have bounded Lipschitz constants. Nowadays, most Normalizing Flows are bi-Lipschitz by design or by training to limit numerical errors (among other things). In this paper, we discuss the expressivity of bi-Lipschitz Normalizing Flows and identify several target distributions that are difficult to approximate using such models. Then, we characterize the expressivity of bi-Lipschitz Normalizing Flows by giving several lower bounds on the Total Variation distance between these particularly unfavorable distributions and their best possible approximation. Finally, we discuss potential remedies which include using more complex latent distributions.
Expressivity of Bi-Lipschitz Normalizing Flows: A Score-Based Diffusion Perspective
Center for Industrial Mathematics, University of Bremen（工业数学中心，不莱梅大学）；Department of Theoretical Physics and Applied Mathematics, University of Cambridge（理论物理与应用数学系，剑桥大学）
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, …
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.
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…
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…
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 have grown more popular over the last few years; however, they continue to be computationally expensive, making them difficult to be accepted into the broader machine learning community. In this paper, we introduce a simple one-dimensional one-layer network that has closed form Lipschitz constants; using this, we introduce a new Exact-Lipschitz Flow (ELF) that combines the ease of sampling from residual flows with the strong performance of autoregressive flows. Further, we show that ELF is provably a universal density approximator, more computationally and parameter efficient…
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.
Normalizing flows, which learn a distribution by transforming the data to samples from a Gaussian base distribution, have proven powerful density approximations. But their expressive power is limited by this choice of the base distribution. We, therefore, propose to generalize the base distribution to a more elaborate copula distribution to capture the properties of the target distribution more accurately. In a first empirical analysis, we demonstrate that this replacement can dramatically improve the vanilla normalizing flows in terms of flexibility, stability, and effectivity for heavy-taile…
Normalizing flows are bijective mappings between inputs and latent representations with a fully factorized distribution. They are very attractive due to exact likelihood valuation and efficient sampling. However, their effective capacity is often insufficient since the bijectivity constraint limits the model width. We address this issue by incrementally padding intermediate representations with noise. We precondition the noise in accordance with previous invertible units, which we describe as cross-unit coupling. Our invertible glow-like modules increase the model expressivity by fusing a dens…
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 …
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…
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…
Medical image segmentation is inherently an ambiguous task due to factors such as partial volumes and variations in anatomical definitions. While in most cases the segmentation uncertainty is around the border of structures of interest, there can also be considerable inter-rater differences. The class of conditional variational autoencoders (cVAE) offers a principled approach to inferring distributions over plausible segmentations that are conditioned on input images. Segmentation uncertainty estimated from samples of such distributions can be more informative than using pixel level probabilit…
MILA（蒙特利尔学习算法研究所）；University of Montreal（蒙特利尔大学）；Element AI（元素人工智能公司）
Normalizing flows and autoregressive models have been successfully combined to produce state-of-the-art results in density estimation, via Masked Autoregressive Flows (MAF), and to accelerate state-of-the-art WaveNet-based speech synthesis to 20x faster than real-time, via Inverse Autoregressive Flows (IAF). We unify and generalize these approaches, replacing the (conditionally) affine univariate transformations of MAF/IAF with a more general class of invertible univariate transformations expressed as monotonic neural networks. We demonstrate that the proposed neural autoregressive flows (NAF)…
ENS Paris Saclay（巴黎萨克雷高等师范学院）；CNRS（法国国家科学研究中心）；Université Paris Cité（巴黎西岱大学）；Institut Universitaire de France（法兰西大学研究院）；ENS Ulm（巴黎高等师范学院）
Many generative models synthesize data by transforming a standard Gaussian random variable using a deterministic neural network. Among these models are the Variational Autoencoders and the Generative Adversarial Networks. In this work, we call them "push-forward" models and study their expressivity. We show that the Lipschitz constant of these generative networks has to be large in order to fit multimodal distributions. More precisely, we show that the total variation distance and the Kullback-Leibler divergence between the generated and the data distribution are bounded from below by a consta…
Non-Vacuous Certification of Transport MCMC via Oscillation-Controlled Normalizing Flows
China Sanya Science and Education Innovation Park, Wuhan University of Technology, Sanya 572025, China（中国三亚科技教育创新园，武汉理工大学，三亚572025，中国）；School of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, China（武汉理工大学土木工程与建筑学院，武汉430070，中国）
Iterative Gaussianization is a fixed-point iteration procedure that can transform any continuous random vector into a Gaussian one. Based on iterative Gaussianization, we propose a new type of normalizing flow model that enables both efficient computation of likelihoods and efficient inversion for sample generation. We demonstrate that these models, named Gaussianization flows, are universal approximators for continuous probability distributions under some regularity conditions. Because of this guaranteed expressivity, they can capture multimodal target distributions without compromising the e…
CMAP, Ecole Polytechnique, Universite Paris-Saclay（巴黎萨克雷大学综合理工学院CMAP实验室）；CDISE, Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所CDISE实验室）；Ecole Normale Supérieure Paris-Saclay（巴黎萨克雷高等师范学院）
In this contribution, we propose a new computationally efficient method to combine Variational Inference (VI) with Markov Chain Monte Carlo (MCMC). This approach can be used with generic MCMC kernels, but is especially well suited to \textit{MetFlow}, a novel family of MCMC algorithms we introduce, in which proposals are obtained using Normalizing Flows. The marginal distribution produced by such MCMC algorithms is a mixture of flow-based distributions, thus drastically increasing the expressivity of the variational family. Unlike previous methods following this direction, our approach is amen…
Frequently, population studies feature pyramidally-organized data represented using Hierarchical Bayesian Models (HBM) enriched with plates. These models can become prohibitively large in settings such as neuroimaging, where a sample is composed of a functional MRI signal measured on 300 brain locations, across 4 measurement sessions, and 30 subjects, resulting in around 1 million latent parameters.Such high dimensionality hampers the usage of modern, expressive flow-based techniques.To infer parameter posterior distributions in this challenging class of problems, we designed a novel methodolo…
Towards Flexibility and Interpretability of Gaussian Process State-Space Model
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Future Network of Intelligence Institute（未来网络智能研究院）；Universidad Autónoma de Madrid（马德里自治大学）；Cognizant AI（高知特人工智能）
Riemannian AmbientFlow: Towards Simultaneous Manifold Learning and Generative Modeling from Corrupted Data
Department of Mathematics, University of California, Los Angeles（数学系，加州大学洛杉矶分校）；Department of Statistics and Data Science, University of California, Los Angeles（统计与数据科学系，加州大学洛杉矶分校）
