In this paper, we present a new class of invertible transformations with an application to flow-based generative models. We indicate that many well-known invertible transformations in reversible logic and reversible neural networks could be derived from our proposition. Next, we propose two new coupling layers that are important building blocks of flow-based generative models. In the experiments on digit data, we present how these new coupling layers could be used in Integer Discrete Flows (IDF), and that they achieve better results than standard coupling layers used in IDF and RealNVP.
University of Toronto（多伦多大学）；University of Bremen（不来梅大学）；Vector Institute（向量研究所）
Flow-based generative models parameterize probability distributions through an invertible transformation and can be trained by maximum likelihood. Invertible residual networks provide a flexible family of transformations where only Lipschitz conditions rather than strict architectural constraints are needed for enforcing invertibility. However, prior work trained invertible residual networks for density estimation by relying on biased log-density estimates whose bias increased with the network's expressiveness. We give a tractable unbiased estimate of the log density using a "Russian roulette"…
Avast Software s.r.o（Avast软件有限责任公司）；Graz University of Technology（格拉茨工业大学）
In this work, we propose Sum-Product-Transform Networks (SPTN), an extension of sum-product networks that uses invertible transformations as additional internal nodes. The type and placement of transformations determine properties of the resulting SPTN with many interesting special cases. Importantly, SPTN with Gaussian leaves and affine transformations pose the same inference task tractable that can be computed efficiently in SPNs. We propose to store affine transformations in their SVD decompositions using an efficient parametrization of unitary matrices by a set of Givens rotations. Last bu…
Flow-based generative models (Dinh et al., 2014) are conceptually attractive due to tractability of the exact log-likelihood, tractability of exact latent-variable inference, and parallelizability of both training and synthesis. In this paper we propose Glow, a simple type of generative flow using an invertible 1x1 convolution. Using our method we demonstrate a significant improvement in log-likelihood on standard benchmarks. Perhaps most strikingly, we demonstrate that a generative model optimized towards the plain log-likelihood objective is capable of efficient realistic-looking synthesis 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…
University of Science and Technology of China（中国科学技术大学）
Generative flows models enjoy the properties of tractable exact likelihood and efficient sampling, which are composed of a sequence of invertible functions. In this paper, we incorporate matrix exponential into generative flows. Matrix exponential is a map from matrices to invertible matrices, this property is suitable for generative flows. Based on matrix exponential, we propose matrix exponential coupling layers that are a general case of affine coupling layers and matrix exponential invertible 1 x 1 convolutions that do not collapse during training. And we modify the networks architecture t…
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…
An invertible generative model for forward and inverse problems
Centrum Wiskunde en Informatica, Amsterdam, The Netherlands（荷兰阿姆斯特丹数学与信息学研究中心）；Utrecht University, Utrecht, The Netherlands（荷兰乌得勒支大学）；University of Twente, Enschede, The Netherlands（荷兰埃因霍温大学）
OnePlanet Research Center（壹星球研究中心）；imec-the Netherlands（荷兰微电子研究中心）；Donders Institute for Brain, Cognition and Behaviour（唐德斯脑、认知与行为研究所）；Radboud University（拉德堡德大学）；UvA-Bosch Delta Lab（阿姆斯特丹大学-博世三角洲实验室）
In this work, we provide a deterministic alternative to the stochastic variational training of generative autoencoders. We refer to these new generative autoencoders as AutoEncoders within Flows (AEF), since the encoder and decoder are defined as affine layers of an overall invertible architecture. This results in a deterministic encoding of the data, as opposed to the stochastic encoding of VAEs. The paper introduces two related families of AEFs. The first family relies on a partition of the ambient space and is trained by exact maximum-likelihood. The second family exploits a deterministic e…
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…
ETH Zürich（苏黎世联邦理工学院）；SMU（新加坡管理大学）；KU Leuven（鲁汶大学）
Flow-based generative models have shown an excellent ability to explicitly learn the probability density function of data via a sequence of invertible transformations. Yet, learning attentions in generative flows remains understudied, while it has made breakthroughs in other domains. To fill the gap, this paper introduces two types of invertible attention mechanisms, i.e., map-based and transformer-based attentions, for both unconditional and conditional generative flows. The key idea is to exploit a masked scheme of these two attentions to learn long-range data dependencies in the context of …
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所HIIT）；Aalto University（阿尔托大学）
While MCMC methods have become a main work-horse for Bayesian inference, scaling them to large distributed datasets is still a challenge. Embarrassingly parallel MCMC strategies take a divide-and-conquer stance to achieve this by writing the target posterior as a product of subposteriors, running MCMC for each of them in parallel and subsequently combining the results. The challenge then lies in devising efficient aggregation strategies. Current strategies trade-off between approximation quality, and costs of communication and computation. In this work, we introduce a novel method that address…
We propose a Multiscale Invertible Generative Network (MsIGN) and associated training algorithm that leverages multiscale structure to solve high-dimensional Bayesian inference. To address the curse of dimensionality, MsIGN exploits the low-dimensional nature of the posterior, and generates samples from coarse to fine scale (low to high dimension) by iteratively upsampling and refining samples. MsIGN is trained in a multi-stage manner to minimize the Jeffreys divergence, which avoids mode dropping in high-dimensional cases. On two high-dimensional Bayesian inverse problems, we show superior pe…
We propose a new way of constructing invertible neural networks by combining simple building blocks with a novel set of composition rules. This leads to a rich set of invertible architectures, including those similar to ResNets. Inversion is achieved with a locally convergent iterative procedure that is parallelizable and very fast in practice. Additionally, the determinant of the Jacobian can be computed analytically and efficiently, enabling their generative use as flow models. To demonstrate their flexibility, we show that our invertible neural networks are competitive with ResNets on MNIST…
Training and using modern neural-network based latent-variable generative models (like Variational Autoencoders) often require simultaneously training a generative direction along with an inferential(encoding) direction, which approximates the posterior distribution over the latent variables. Thus, the question arises: how complex does the inferential model need to be, in order to be able to accurately model the posterior distribution of a given generative model? In this paper, we identify an important property of the generative map impacting the required size of the encoder. We show that if t…
eBay；Inception Institute of Artificial Intelligence（人工智能初学者研究所）
Deep generative models have been successfully applied to Zero-Shot Learning (ZSL) recently. However, the underlying drawbacks of GANs and VAEs (e.g., the hardness of training with ZSL-oriented regularizers and the limited generation quality) hinder the existing generative ZSL models from fully bypassing the seen-unseen bias. To tackle the above limitations, for the first time, this work incorporates a new family of generative models (i.e., flow-based models) into ZSL. The proposed Invertible Zero-shot Flow (IZF) learns factorized data embeddings (i.e., the semantic factors and the non-semantic…
Carnegie Mellon University（卡内基梅隆大学）；University of Southern California（南加州大学）；University of California, Berkeley（加利福尼亚大学伯克利分校）
In this work, we propose a new generative model that is capable of automatically decoupling global and local representations of images in an entirely unsupervised setting, by embedding a generative flow in the VAE framework to model the decoder. Specifically, the proposed model utilizes the variational auto-encoding framework to learn a (low-dimensional) vector of latent variables to capture the global information of an image, which is fed as a conditional input to a flow-based invertible decoder with architecture borrowed from style transfer literature. Experimental results on standard image …
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…
The University of Tokyo（东京大学）；Preferred Networks, Inc.（优选网络公司）
Statistical generative models for molecular graphs attract attention from many researchers from the fields of bio- and chemo-informatics. Among these models, invertible flow-based approaches are not fully explored yet. In this paper, we propose a powerful invertible flow for molecular graphs, called graph residual flow (GRF). The GRF is based on residual flows, which are known for more flexible and complex non-linear mappings than traditional coupling flows. We theoretically derive non-trivial conditions such that GRF is invertible, and present a way of keeping the entire flows invertible thro…
Tooploox；University of Wrocław（弗罗茨瓦夫大学）；Wrocław University of Science and Technology（弗罗茨瓦夫理工大学）；NavAlgo
This paper focuses on a novel generative approach for 3D point clouds that makes use of invertible flow-based models. The main idea of the method is to treat a point cloud as a probability density in 3D space that is modeled using a cloud-specific neural network. To capture the similarity between point clouds we rely on parameter sharing among networks, with each cloud having only a small embedding vector that defines it. We use invertible flows networks to generate the individual point clouds, and to regularize the embedding vectors. We evaluate the generative capabilities of the model both i…
Quantum Artificial Intelligence Laboratory (QuAIL), NASA Ames Research Centre（美国国家航空航天局艾姆斯研究中心量子人工智能实验室（QuAIL））；KBR（KBR公司）；University College London（伦敦大学学院）；London Centre for Nanotechnology（伦敦纳米技术中心）
Efficient sampling of complex data distributions can be achieved using trained invertible flows (IF), where the model distribution is generated by pushing a simple base distribution through multiple non-linear bijective transformations. However, the iterative nature of the transformations in IFs can limit the approximation to the target distribution. In this paper we seek to mitigate this by implementing RBM-Flow, an IF model whose base distribution is a Restricted Boltzmann Machine (RBM) with a continuous smoothing applied. We show that by using RBM-Flow we are able to improve the quality of …
University of Bremen（不来梅大学）；Vector Institute（向量研究所）；University of Toronto（多伦多大学）
We show that standard ResNet architectures can be made invertible, allowing the same model to be used for classification, density estimation, and generation. Typically, enforcing invertibility requires partitioning dimensions or restricting network architectures. In contrast, our approach only requires adding a simple normalization step during training, already available in standard frameworks. Invertible ResNets define a generative model which can be trained by maximum likelihood on unlabeled data. To compute likelihoods, we introduce a tractable approximation to the Jacobian log-determinant …
Many recent invertible neural architectures are based on coupling block designs where variables are divided in two subsets which serve as inputs of an easily invertible (usually affine) triangular transformation. While such a transformation is invertible, its Jacobian is very sparse and thus may lack expressiveness. This work presents a simple remedy by noting that subdivision and (affine) coupling can be repeated recursively within the resulting subsets, leading to an efficiently invertible block with dense, triangular Jacobian. By formulating our recursive coupling scheme via a hierarchical …
Georgia Institute of Technology（佐治亚理工学院）；Duke University（杜克大学）
Graph prediction problems prevail in data analysis and machine learning. The inverse prediction problem, namely to infer input data from given output labels, is of emerging interest in various applications. In this work, we develop \textit{invertible graph neural network} (iGNN), a deep generative model to tackle the inverse prediction problem on graphs by casting it as a conditional generative task. The proposed model consists of an invertible sub-network that maps one-to-one from data to an intermediate encoded feature, which allows forward prediction by a linear classification sub-network a…
Interdisciplinary Center for Scientific Computing, Universität Heidelberg（海德堡大学跨学科科学计算中心）；Institute for Particle Physics Phenomenology, Department of Physics, Durham University（杜伦大学物理系粒子物理现象学研究所）
Recently pretrained generative models have shown promising results for subsampled Fourier Ptychography (FP) in terms of quality of reconstruction for extremely low sampling rate and high noise. However, one of the significant drawbacks of these pretrained generative priors is their limited representation capabilities. Moreover, training these generative models requires access to a large number of fully-observed clean samples of a particular class of images like faces or digits that is prohibitive to obtain in the context of FP. In this paper, we propose to leverage the power of pretrained inve…
MPI for Intelligent Systems, Tübingen（马克斯·普朗克智能系统研究所，图宾根）；Amazon.com, Inc.（亚马逊公司）
Generation of photo-realistic images, semantic editing and representation learning are a few of many potential applications of high resolution generative models. Recent progress in GANs have established them as an excellent choice for such tasks. However, since they do not provide an inference model, image editing or downstream tasks such as classification can not be done on real images using the GAN latent space. Despite numerous efforts to train an inference model or design an iterative method to invert a pre-trained generator, previous methods are dataset (e.g. human face images) and archit…
Generating molecular graphs with desired chemical properties driven by deep graph generative models provides a very promising way to accelerate drug discovery process. Such graph generative models usually consist of two steps: learning latent representations and generation of molecular graphs. However, to generate novel and chemically-valid molecular graphs from latent representations is very challenging because of the chemical constraints and combinatorial complexity of molecular graphs. In this paper, we propose MoFlow, a flow-based graph generative model to learn invertible mappings between…
University of Wrocław（弗罗茨瓦夫大学）；Tooploox；Wrocław University of Science and Technology（弗罗茨瓦夫理工大学）；Warsaw University of Technology（华沙理工大学）；NavAlgo
In this paper, we propose a simple yet effective method to represent point clouds as sets of samples drawn from a cloud-specific probability distribution. This interpretation matches intrinsic characteristics of point clouds: the number of points and their ordering within a cloud is not important as all points are drawn from the proximity of the object boundary. We postulate to represent each cloud as a parameterized probability distribution defined by a generative neural network. Once trained, such a model provides a natural framework for point cloud manipulation operations, such as aligning …
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；OpenAI
A promising class of generative models maps points from a simple distribution to a complex distribution through an invertible neural network. Likelihood-based training of these models requires restricting their architectures to allow cheap computation of Jacobian determinants. Alternatively, the Jacobian trace can be used if the transformation is specified by an ordinary differential equation. In this paper, we use Hutchinson's trace estimator to give a scalable unbiased estimate of the log-density. The result is a continuous-time invertible generative model with unbiased density estimation an…
Pennsylvania State University（宾夕法尼亚州立大学）；McGill University（麦吉尔大学）；Shanghai Jiao Tong University（上海交通大学）；Chinese Academy of Sciences（中国科学院）
Deep neural networks have proven to be quite effective in a wide variety of machine learning tasks, ranging from improved speech recognition systems to advancing the development of autonomous vehicles. However, despite their superior performance in many applications, these models have been recently shown to be susceptible to a particular type of attack possible through the generation of particular synthetic examples referred to as adversarial samples. These samples are constructed by manipulating real examples from the training data distribution in order to "fool" the original neural model, re…
Language Technologies Institute（语言技术研究所）；School of Computer Science（计算机科学学院）；Carnegie Mellon University（卡内基梅隆大学）
Unsupervised learning of syntactic structure is typically performed using generative models with discrete latent variables and multinomial parameters. In most cases, these models have not leveraged continuous word representations. In this work, we propose a novel generative model that jointly learns discrete syntactic structure and continuous word representations in an unsupervised fashion by cascading an invertible neural network with a structured generative prior. We show that the invertibility condition allows for efficient exact inference and marginal likelihood computation in our model so…
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…
Ben-Gurion University of the Negev（内盖夫本-古里安大学）；NEC Corporation（日本电气股份有限公司）
Model agnostic feature attribution algorithms (such as SHAP and LIME) are ubiquitous techniques for explaining the decisions of complex classification models, such as deep neural networks. However, since complex classification models produce superior performance when trained on low-level (or encoded) features, in many cases, the explanations generated by these algorithms are neither interpretable nor usable by humans. Methods proposed in recent studies that support the generation of human-interpretable explanations are impractical, because they require a fully invertible transformation functio…
A normalizing flow models a complex probability density as an invertible transformation of a simple base density. Flows based on either coupling or autoregressive transforms both offer exact density evaluation and sampling, but rely on the parameterization of an easily invertible elementwise transformation, whose choice determines the flexibility of these models. Building upon recent work, we propose a fully-differentiable module based on monotonic rational-quadratic splines, which enhances the flexibility of both coupling and autoregressive transforms while retaining analytic invertibility. W…
We introduce Invertible Dense Networks (i-DenseNets), a more parameter efficient extension of Residual Flows. The method relies on an analysis of the Lipschitz continuity of the concatenation in DenseNets, where we enforce invertibility of the network by satisfying the Lipschitz constant. Furthermore, we propose a learnable weighted concatenation, which not only improves the model performance but also indicates the importance of the concatenated weighted representation. Additionally, we introduce the Concatenated LipSwish as activation function, for which we show how to enforce the Lipschitz c…
In this paper, we propose a deep invertible hybrid model which integrates discriminative and generative learning at a latent space level for semi-supervised few-shot classification. Various tasks for classifying new species from image data can be modeled as a semi-supervised few-shot classification, which assumes a labeled and unlabeled training examples and a small support set of the target classes. Predicting target classes with a few support examples per class makes the learning task difficult for existing semi-supervised classification methods, including selftraining, which iteratively est…
Yale University（耶鲁大学）；Pennsylvania State University（宾夕法尼亚州立大学）
Owing much to the revolution of information technology, the recent progress of deep learning benefits incredibly from the vastly enhanced access to data available in various digital formats. However, in certain scenarios, people may not want their data being used for training commercial models and thus studied how to attack the learnability of deep learning models. Previous works on learnability attack only consider the goal of preventing unauthorized exploitation on the specific dataset but not the process of restoring the learnability for authorized cases. To tackle this issue, this paper in…
Normalising flows offer a flexible way of modelling continuous probability distributions. We consider expressiveness, fast inversion and exact Jacobian determinant as three desirable properties a normalising flow should possess. However, few flow models have been able to strike a good balance among all these properties. Realising that the integral of a convex sum of sinusoidal functions squared leads to a bijective residual transformation, we propose Sinusoidal Flow, a new type of normalising flows that inherits the expressive power and triangular Jacobian from fully autoregressive flows while…
In this work, we develop a new approach to generative density estimation for exchangeable, non-i.i.d. data. The proposed framework, FlowScan, combines invertible flow transformations with a sorted scan to flexibly model the data while preserving exchangeability. Unlike most existing methods, FlowScan exploits the intradependencies within sets to learn both global and local structure. FlowScan represents the first approach that is able to apply sequential methods to exchangeable density estimation without resorting to averaging over all possible permutations. We achieve new state-of-the-art per…
International Institute of Information Technology Hyderabad（海得拉巴国际信息技术学院）；École Normale Supérieure (ENS) Paris-Saclay（巴黎萨克雷高等师范学院）
Normalizing flows are an essential alternative to GANs for generative modelling, which can be optimized directly on the maximum likelihood of the dataset. They also allow computation of the exact latent vector corresponding to an image since they are composed of invertible transformations. However, the requirement of invertibility of the transformation prevents standard and expressive neural network models such as CNNs from being directly used. Emergent convolutions were proposed to construct an invertible 3$\times$3 CNN layer using a pair of masked CNN layers, making them inefficient. We stud…
Monotonic neural networks have recently been proposed as a way to define invertible transformations. These transformations can be combined into powerful autoregressive flows that have been shown to be universal approximators of continuous probability distributions. Architectures that ensure monotonicity typically enforce constraints on weights and activation functions, which enables invertibility but leads to a cap on the expressiveness of the resulting transformations. In this work, we propose the Unconstrained Monotonic Neural Network (UMNN) architecture based on the insight that a function …
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…
Tsinghua University（清华大学）；Peng Cheng Laboratory（鹏城实验室）；Renmin University of China（中国人民大学）
By applying entropy codecs with learned data distributions, neural compressors have significantly outperformed traditional codecs in terms of compression ratio. However, the high inference latency of neural networks hinders the deployment of neural compressors in practical applications. In this work, we propose Integer-only Discrete Flows (IODF), an efficient neural compressor with integer-only arithmetic. Our work is built upon integer discrete flows, which consists of invertible transformations between discrete random variables. We propose efficient invertible transformations with integer-on…
Acoustics Research Institute, Austrian Academy of Sciences（奥地利科学院声学研究所）；Swiss Data Science Center, ETH Zürich（苏黎世联邦理工学院瑞士数据科学中心）
Time-frequency (TF) representations provide powerful and intuitive features for the analysis of time series such as audio. But still, generative modeling of audio in the TF domain is a subtle matter. Consequently, neural audio synthesis widely relies on directly modeling the waveform and previous attempts at unconditionally synthesizing audio from neurally generated invertible TF features still struggle to produce audio at satisfying quality. In this article, focusing on the short-time Fourier transform, we discuss the challenges that arise in audio synthesis based on generated invertible TF f…
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…
It is nontrivial to store rapidly growing big data nowadays, which demands high-performance lossless compression techniques. Likelihood-based generative models have witnessed their success on lossless compression, where flow based models are desirable in allowing exact data likelihood optimisation with bijective mappings. However, common continuous flows are in contradiction with the discreteness of coding schemes, which requires either 1) imposing strict constraints on flow models that degrades the performance or 2) coding numerous bijective mapping errors which reduces the efficiency. In thi…
