This paper introduces the Variational Determinant Estimator (VDE), a variational extension of the recently proposed determinant estimator discovered by arXiv:2005.06553v2. Our estimator significantly reduces the variance even for low sample sizes by combining (importance-weighted) variational inference and a family of normalizing flows which allow density estimation on hyperspheres. In the ideal case of a tight variational bound, the VDE becomes a zero variance estimator, and a single sample is sufficient for an exact (log) determinant estimate.
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…
Efficient gradient computation of the Jacobian determinant term is a core problem in many machine learning settings, and especially so in the normalizing flow framework. Most proposed flow models therefore either restrict to a function class with easy evaluation of the Jacobian determinant, or an efficient estimator thereof. However, these restrictions limit the performance of such density models, frequently requiring significant depth to reach desired performance levels. In this work, we propose Self Normalizing Flows, a flexible framework for training normalizing flows by replacing expensive…
Emory University（埃默里大学）；University of California, Los Angeles（加州大学洛杉矶分校）
A normalizing flow is an invertible mapping between an arbitrary probability distribution and a standard normal distribution; it can be used for density estimation and statistical inference. Computing the flow follows the change of variables formula and thus requires invertibility of the mapping and an efficient way to compute the determinant of its Jacobian. To satisfy these requirements, normalizing flows typically consist of carefully chosen components. Continuous normalizing flows (CNFs) are mappings obtained by solving a neural ordinary differential equation (ODE). The neural ODE's dynami…
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）
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…
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 are objects used for modeling complicated probability density functions, and have attracted considerable interest in recent years. Many flexible families of normalizing flows have been developed. However, the focus to date has largely been on normalizing flows on Euclidean domains; while normalizing flows have been developed for spherical and other non-Euclidean domains, these are generally less flexible than their Euclidean counterparts. To address this shortcoming, in this work we introduce a mixture-of-normalizing-flows model to construct complicated probability density fu…
Colorado School of Mines（科罗拉多矿业学院）；University of California, Los Angeles（加州大学洛杉矶分校）
A normalizing flow (NF) is a mapping that transforms a chosen probability distribution to a normal distribution. Such flows are a common technique used for data generation and density estimation in machine learning and data science. The density estimate obtained with a NF requires a change of variables formula that involves the computation of the Jacobian determinant of the NF transformation. In order to tractably compute this determinant, continuous normalizing flows (CNF) estimate the mapping and its Jacobian determinant using a neural ODE. Optimal transport (OT) theory has been successfully…
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…
Waveflow: boundary-conditioned normalizing flows applied to fermionic wavefunctions
University of Toronto（多伦多大学）；Vector Institute for Artificial Intelligence（向量人工智能研究所）；Canadian Institute for Advanced Research (CIFAR)（加拿大高等研究院）；Acceleration Consortium（加速联盟）
JFlow: Model-Independent Spherical Jeans Analysis using Equivariant Continuous Normalizing Flows
Particle Theory and Cosmology Group, Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS)（基本科学研究院（IBS）粒子理论与宇宙学组，宇宙理论物理中心）；Department of Physics and Astronomy, Rutgers, The State University of New Jersey（新泽西州立大学罗格斯大学物理与天文学系）；National Institute of Technology, Sendai College（Sendai学院国立技术研究所）；Astronomical Institute, Tohoku University（东北大学天文研究所）；ICRR, The University of Tokyo（东京大学ICRR）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo（东京大学Kavli研究所（WPI），东京大学先进研究所，东京大学）；Theory Center, IPNS, KEK（IPNS理论中心，KEK）；Graduate University for Advanced Studies (SOKENDAI)（高级研究大学（SOKENDAI））
Normalizing flows for lattice gauge theory in arbitrary space-time dimension
Massachusetts Institute of Technology（麻省理工学院）；New York University（纽约大学）；DeepMind（深度思维）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of Bern（伯尔尼大学）
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…
Stellenbosch University（斯泰伦博斯大学）；National Institute for Theoretical and Computational Sciences（国家理论与计算科学研究所）
Graphical flows add further structure to normalizing flows by encoding non-trivial variable dependencies. Previous graphical flow models have focused primarily on a single flow direction: the normalizing direction for density estimation, or the generative direction for inference. However, to use a single flow to perform tasks in both directions, the model must exhibit stable and efficient flow inversion. This work introduces graphical residual flows, a graphical flow based on invertible residual networks. Our approach to incorporating dependency information in the flow, means that we are able …
In addition to recent developments in computing speed and memory, methodological advances have contributed to significant gains in the performance of stochastic simulation. In this paper, we focus on variance reduction for matrix computations via matrix factorization. We provide insights into existing variance reduction methods for estimating the entries of large matrices. Popular methods do not exploit the reduction in variance that is possible when the matrix is factorized. We show how computing the square root factorization of the matrix can achieve in some important cases arbitrarily bette…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Max Planck Institute for Biological Cybernetics（马克斯·普朗克生物控制论研究所）；Université Paris-Saclay（巴黎萨克雷大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；University of Helsinki（赫尔辛基大学）
Learning expressive probabilistic models correctly describing the data is a ubiquitous problem in machine learning. A popular approach for solving it is mapping the observations into a representation space with a simple joint distribution, which can typically be written as a product of its marginals -- thus drawing a connection with the field of nonlinear independent component analysis. Deep density models have been widely used for this task, but their maximum likelihood based training requires estimating the log-determinant of the Jacobian and is computationally expensive, thus imposing a tra…
While normalizing flows have led to significant advances in modeling high-dimensional continuous distributions, their applicability to discrete distributions remains unknown. In this paper, we show that flows can in fact be extended to discrete events---and under a simple change-of-variables formula not requiring log-determinant-Jacobian computations. Discrete flows have numerous applications. We consider two flow architectures: discrete autoregressive flows that enable bidirectionality, allowing, for example, tokens in text to depend on both left-to-right and right-to-left contexts in an exac…
Carnegie Mellon University（卡内基梅隆大学）；International Digital Economy Academy（国际数字经济研究院）；Johns Hopkins University（约翰斯·霍普金斯大学）；University of California, Berkeley（加州大学伯克利分校）
The principle of Maximal Coding Rate Reduction (MCR$^2$) has recently been proposed as a training objective for learning discriminative low-dimensional structures intrinsic to high-dimensional data to allow for more robust training than standard approaches, such as cross-entropy minimization. However, despite the advantages that have been shown for MCR$^2$ training, MCR$^2$ suffers from a significant computational cost due to the need to evaluate and differentiate a significant number of log-determinant terms that grows linearly with the number of classes. By taking advantage of variational fo…
UvA-Bosch Delta Lab（阿姆斯特丹大学-博世德尔塔实验室）；University of Amsterdam（阿姆斯特丹大学）；Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）
This paper introduces a new method to build linear flows, by taking the exponential of a linear transformation. This linear transformation does not need to be invertible itself, and the exponential has the following desirable properties: it is guaranteed to be invertible, its inverse is straightforward to compute and the log Jacobian determinant is equal to the trace of the linear transformation. An important insight is that the exponential can be computed implicitly, which allows the use of convolutional layers. Using this insight, we develop new invertible transformations named convolution e…
Normalizing Flows (NF) are Generative models which transform a simple prior distribution into the desired target. They however require the design of an invertible mapping whose Jacobian determinant has to be computable. Recently introduced, Neural Hamiltonian Flows (NHF) are Hamiltonian dynamics-based flows, which are continuous, volume-preserving and invertible and thus make for natural candidates for robust NF architectures. In particular, their similarity to classical Mechanics could lead to easier interpretability of the learned mapping. In this paper, we show that the current NHF architec…
Flow Perturbation++: Multi-Step Unbiased Jacobian Estimation for High-Dimensional Boltzmann Sampling
State Key Laboratory of Information Photonics（信息光子学与光通信国家重点实验室）；Optical Communications, Beijing University of Posts（光通信，北京邮电大学）；School of Physical Science（物理科学学院）；Technology, Beijing University of Posts（技术学院，北京邮电大学）
A trivializing map is a field transformation whose Jacobian determinant exactly cancels the interaction terms in the action, providing a representation of the theory in terms of a deterministic transformation of a distribution from which sampling is trivial. Recently, a proof-of-principle study by Albergo, Kanwar and Shanahan [arXiv:1904.12072] demonstrated that approximations of trivializing maps can be `machine-learned' by a class of invertible, differentiable neural models called \textit{normalizing flows}. By ensuring that the Jacobian determinant can be computed efficiently, asymptoticall…
Weak Generative Sampler to Efficiently Sample Invariant Distribution of Stochastic Differential Equation
Department of Data Science, City University of Hong Kong（数据科学系，香港城市大学）；Institute of Natural Sciences and School of Mathematical Sciences, Shanghai Jiao Tong University（自然科学院和数学科学学院，上海交通大学）；Department of Mathematics and Department of Data Science, City University of Hong Kong（数学系和数据科学系，香港城市大学）
Cornell Tech and Cornell University（康奈尔科技校区和康奈尔大学）
Numerous applications of machine learning involve representing probability distributions over high-dimensional data. We propose autoregressive quantile flows, a flexible class of normalizing flow models trained using a novel objective based on proper scoring rules. Our objective does not require calculating computationally expensive determinants of Jacobians during training and supports new types of neural architectures, such as neural autoregressive flows from which it is easy to sample. We leverage these models in quantile flow regression, an approach that parameterizes predictive conditiona…
