VAE-MS: An Asymmetric Variational Autoencoder for Mutational Signature Extraction
Center for Clinical Data Science（临床数据科学中心）；Aalborg University（奥胡斯大学）；Aalborg University Hospital（奥胡斯大学医院）；Nuffield Department of Surgical Sciences（外科科学努尔菲尔德部门）；University of Oxford（牛津大学）；Nuffield Department of Women’s and Reproductive Health（妇女和生殖健康努尔菲尔德部门）
Ida Egendal, Rasmus Froberg Brøndum, Dan J Woodcock, Christopher Yau, Martin Bøgsted
Mutational signature analysis has emerged as a powerful method for uncovering the underlying biological processes driving cancer development. However, the signature extraction process, typically performed using non-negative matrix factorization (NMF), often lacks reliability and clinical applicability. To address these limitations, several solutions have been introduced, including the use of neural networks to achieve more accurate estimates and probabilistic methods to better capture natural variation in the data. In this work, we introduce a Variational Autoencoder for Mutational Signatures (VAE-MS), a novel model that leverages both an asymmetric architecture and probabilistic methods for the extraction of mutational signatures. VAE-MS is compared to with three state-of-the-art models for mutational signature extraction: SigProfilerExtractor, the NMF-based gold standard; MUSE-XAE, an autoencoder that employs an asymmetric design without probabilistic components; and SigneR, a Bayesian NMF model, to illustrate the strength in combining a nonlinear extraction with a probabilistic model. In the ability to reconstruct input data and generalize to unseen data, models with probabilistic components (VAE-MS, SigneR) dramatically outperformed models without (SigProfilerExtractor, MUSE-XAE). The NMF-baed models (SigneR, SigProfilerExtractor) had the most accurate reconstructions in simulated data, while VAE-MS reconstructed more accurately on real cancer data. Upon evaluating the ability to extract signatures consistently, no model exhibited a clear advantage over the others. Software for VAE-MS is available at https://github.com/CLINDA-AAU/VAE-MS.
University of Florida（佛罗里达大学）；NEC Laboratories Europe（NEC欧洲实验室）
Although substantial efforts have been made to learn disentangled representations under the variational autoencoder (VAE) framework, the fundamental properties to the dynamics of learning of most VAE models still remain unknown and under-investigated. In this work, we first propose a novel learning objective, termed the principle-of-relevant-information variational autoencoder (PRI-VAE), to learn disentangled representations. We then present an information-theoretic perspective to analyze existing VAE models by inspecting the evolution of some critical information-theoretic quantities across t…
This work gives an in-depth derivation of the trainable evidence lower bound obtained from the marginal joint log-Likelihood with the goal of training a Multi-Modal Variational Autoencoder (M$^2$VAE).
Among likelihood-based approaches for deep generative modelling, variational autoencoders (VAEs) offer scalable amortized posterior inference and fast sampling. However, VAEs are also more and more outperformed by competing models such as normalizing flows (NFs), deep-energy models, or the new denoising diffusion probabilistic models (DDPMs). In this preliminary work, we improve VAEs by demonstrating how DDPMs can be used for modelling the prior distribution of the latent variables. The diffusion prior model improves upon Gaussian priors of classical VAEs and is competitive with NF-based prior…
University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦·图灵研究所）
Variational autoencoders (VAEs) are a powerful class of deep generative latent variable model for unsupervised representation learning on high-dimensional data. To ensure computational tractability, VAEs are often implemented with a univariate standard Gaussian prior and a mean-field Gaussian variational posterior distribution. This results in a vector-valued latent variables that are agnostic to the original data structure which might be highly correlated across and within multiple dimensions. We propose a tensor-variate extension to the VAE framework, the tensor-variate Gaussian process prio…
Variational autoencoders (VAEs) have ushered in a new era of unsupervised learning methods for complex distributions. Although these techniques are elegant in their approach, they are typically not useful for representation learning. In this work, we propose a simple yet powerful class of VAEs that simultaneously result in meaningful learned representations. Our solution is to combine traditional VAEs with mutual information maximization, with the goal to enhance amortized inference in VAEs using Information Theoretic techniques. We call this approach InfoMax-VAE, and such an approach can sign…
RETINA-AI Health, Inc.（RETINA-AI健康公司）；MD Anderson Cancer Center（MD安德森癌症中心）
In Bayesian machine learning, the posterior distribution is typically computationally intractable, hence variational inference is often required. In this approach, an evidence lower bound on the log likelihood of data is maximized during training. Variational Autoencoders (VAE) are one important example where variational inference is utilized. In this tutorial, we derive the variational lower bound loss function of the standard variational autoencoder. We do so in the instance of a gaussian latent prior and gaussian approximate posterior, under which assumptions the Kullback-Leibler term in th…
We present VAEL, a neuro-symbolic generative model integrating variational autoencoders (VAE) with the reasoning capabilities of probabilistic logic (L) programming. Besides standard latent subsymbolic variables, our model exploits a probabilistic logic program to define a further structured representation, which is used for logical reasoning. The entire process is end-to-end differentiable. Once trained, VAEL can solve new unseen generation tasks by (i) leveraging the previously acquired knowledge encoded in the neural component and (ii) exploiting new logical programs on the structured laten…
University of Aberdeen（阿伯丁大学）；University of Sheffield（谢菲尔德大学）
Variational Autoencoder (VAE) is a powerful method for learning representations of high-dimensional data. However, VAEs can suffer from an issue known as latent variable collapse (or KL loss vanishing), where the posterior collapses to the prior and the model will ignore the latent codes in generative tasks. Such an issue is particularly prevalent when employing VAE-RNN architectures for text modelling (Bowman et al., 2016). In this paper, we present a simple architecture called holistic regularisation VAE (HR-VAE), which can effectively avoid latent variable collapse. Compared to existing VAE…
Pohang University of Science and Technology（浦项科技大学）
The variational autoencoder (VAE) is a generative model with continuous latent variables where a pair of probabilistic encoder (bottom-up) and decoder (top-down) is jointly learned by stochastic gradient variational Bayes. We first elaborate Gaussian VAE, approximating the local covariance matrix of the decoder as an outer product of the principal direction at a position determined by a sample drawn from Gaussian distribution. We show that this model, referred to as VAE-ROC, better captures the data manifold, compared to the standard Gaussian VAE where independent multivariate Gaussian was use…
Université Pierre et Marie Curie（皮埃尔和玛丽·居里大学）；Sony CSL（索尼计算机科学研究所）；École Polytechnique（巴黎综合理工学院）
VAEs (Variational AutoEncoders) have proved to be powerful in the context of density modeling and have been used in a variety of contexts for creative purposes. In many settings, the data we model possesses continuous attributes that we would like to take into account at generation time. We propose in this paper GLSR-VAE, a Geodesic Latent Space Regularization for the Variational AutoEncoder architecture and its generalizations which allows a fine control on the embedding of the data into the latent space. When augmenting the VAE loss with this regularization, changes in the learned latent spa…
Although deep generative models such as Defense-GAN and Defense-VAE have made significant progress in terms of adversarial defenses of image classification neural networks, several methods have been found to circumvent these defenses. Based on Defense-VAE, in our research we introduce several methods to improve the robustness of defense models. The methods introduced in this paper are straight forward yet show promise over the vanilla Defense-VAE. With extensive experiments on MNIST data set, we have demonstrated the effectiveness of our algorithms against different attacks. Our experiments al…
Modern machine learning systems achieve great success when trained on large datasets. However, these datasets usually contain sensitive information (e.g. medical records, face images), leading to serious privacy concerns. Differentially private generative models (DPGMs) emerge as a solution to circumvent such privacy concerns by generating privatized sensitive data. Similar to other differentially private (DP) learners, the major challenge for DPGM is also how to achieve a subtle balance between utility and privacy. We propose DP$^2$-VAE, a novel training mechanism for variational autoencoders…
In this paper, we propose a Conditioned Variational Autoencoder (C-VAE) for constrained top-N item recommendation where the recommended items must satisfy a given condition. The proposed model architecture is similar to a standard VAE in which the condition vector is fed into the encoder. The constrained ranking is learned during training thanks to a new reconstruction loss that takes the input condition into account. We show that our model generalizes the state-of-the-art Mult-VAE collaborative filtering model. Moreover, we provide insights on what C-VAE learns in the latent space, providing …
When training a variational autoencoder (VAE) on a given dataset, determining the optimal number of latent variables is mostly done by grid search: a costly process in terms of computational time and carbon footprint. In this paper, we explore the intrinsic dimension estimation (IDE) of the data and latent representations learned by VAEs. We show that the discrepancies between the IDE of the mean and sampled representations of a VAE after only a few steps of training reveal the presence of passive variables in the latent space, which, in well-behaved VAEs, indicates a superfluous number of dim…
In this paper, we provide an information-theoretic interpretation of the Vector Quantized-Variational Autoencoder (VQ-VAE). We show that the loss function of the original VQ-VAE can be derived from the variational deterministic information bottleneck (VDIB) principle. On the other hand, the VQ-VAE trained by the Expectation Maximization (EM) algorithm can be viewed as an approximation to the variational information bottleneck(VIB) principle.
Harbin Institute of Technology（哈尔滨工业大学）；Beijing University of Civil Engineering and Architecture（北京建筑大学）
Being one of the most popular generative framework, variational autoencoders(VAE) are known to suffer from a phenomenon termed posterior collapse, i.e. the latent variational distributions collapse to the prior, especially when a strong decoder network is used. In this work, we analyze the latent representation of collapsed VAEs, and proposed a novel model, neighbor embedding VAE(NE-VAE), which explicitly constraints the encoder to encode inputs close in the input space to be close in the latent space. We observed that for VAE variants that report similar ELBO, KL divergence or even mutual inf…
Energy-based models (EBMs) have recently been successful in representing complex distributions of small images. However, sampling from them requires expensive Markov chain Monte Carlo (MCMC) iterations that mix slowly in high dimensional pixel space. Unlike EBMs, variational autoencoders (VAEs) generate samples quickly and are equipped with a latent space that enables fast traversal of the data manifold. However, VAEs tend to assign high probability density to regions in data space outside the actual data distribution and often fail at generating sharp images. In this paper, we propose VAEBM, …
In just three years, Variational Autoencoders (VAEs) have emerged as one of the most popular approaches to unsupervised learning of complicated distributions. VAEs are appealing because they are built on top of standard function approximators (neural networks), and can be trained with stochastic gradient descent. VAEs have already shown promise in generating many kinds of complicated data, including handwritten digits, faces, house numbers, CIFAR images, physical models of scenes, segmentation, and predicting the future from static images. This tutorial introduces the intuitions behind VAEs, e…
University of Oxford（牛津大学）；University College London（伦敦大学学院）；Alan Turing Institute（艾伦·图灵研究所）
We introduce an approach for training Variational Autoencoders (VAEs) that are certifiably robust to adversarial attack. Specifically, we first derive actionable bounds on the minimal size of an input perturbation required to change a VAE's reconstruction by more than an allowed amount, with these bounds depending on certain key parameters such as the Lipschitz constants of the encoder and decoder. We then show how these parameters can be controlled, thereby providing a mechanism to ensure \textit{a priori} that a VAE will attain a desired level of robustness. Moreover, we extend this to a com…
University of Sheffield（谢菲尔德大学）；Utrecht University（乌得勒支大学）
The Variational Autoencoder (VAE) is a popular and powerful model applied to text modelling to generate diverse sentences. However, an issue known as posterior collapse (or KL loss vanishing) happens when the VAE is used in text modelling, where the approximate posterior collapses to the prior, and the model will totally ignore the latent variables and be degraded to a plain language model during text generation. Such an issue is particularly prevalent when RNN-based VAE models are employed for text modelling. In this paper, we propose a simple, generic architecture called Timestep-Wise Regula…
Recently, an audio-visual speech generative model based on variational autoencoder (VAE) has been proposed, which is combined with a nonnegative matrix factorization (NMF) model for noise variance to perform unsupervised speech enhancement. When visual data is clean, speech enhancement with audio-visual VAE shows a better performance than with audio-only VAE, which is trained on audio-only data. However, audio-visual VAE is not robust against noisy visual data, e.g., when for some video frames, speaker face is not frontal or lips region is occluded. In this paper, we propose a robust unsupervi…
We present a preliminary study on an end-to-end variational autoencoder (VAE) for sound morphing. Two VAE variants are compared: VAE with dilation layers (DC-VAE) and VAE only with regular convolutional layers (CC-VAE). We combine the following loss functions: 1) the time-domain mean-squared error for reconstructing the input signal, 2) the Kullback-Leibler divergence to the standard normal distribution in the bottleneck layer, and 3) the classification loss calculated from the bottleneck representation. On a database of spoken digits, we use 1-nearest neighbor classification to show that the …
University of Oxford（牛津大学）；UCL（伦敦大学学院）；The Alan Turing Institute（艾伦·图灵研究所）
In this work we study Variational Autoencoders (VAEs) from the perspective of harmonic analysis. By viewing a VAE's latent space as a Gaussian Space, a variety of measure space, we derive a series of results that show that the encoder variance of a VAE controls the frequency content of the functions parameterised by the VAE encoder and decoder neural networks. In particular we demonstrate that larger encoder variances reduce the high frequency content of these functions. Our analysis allows us to show that increasing this variance effectively induces a soft Lipschitz constraint on the decoder …
Aalto University（阿尔托大学）；University of Helsinki（赫尔辛基大学）；Bayer Pharmaceuticals（拜耳制药）
Conditional variational autoencoders (CVAEs) are versatile deep generative models that extend the standard VAE framework by conditioning the generative model with auxiliary covariates. The original CVAE model assumes that the data samples are independent, whereas more recent conditional VAE models, such as the Gaussian process (GP) prior VAEs, can account for complex correlation structures across all data samples. While several methods have been proposed to learn standard VAEs from partially observed datasets, these methods fall short for conditional VAEs. In this work, we propose a method to …
Graph autoencoders (AE) and variational autoencoders (VAE) are powerful node embedding methods, but suffer from scalability issues. In this paper, we introduce FastGAE, a general framework to scale graph AE and VAE to large graphs with millions of nodes and edges. Our strategy, based on an effective stochastic subgraph decoding scheme, significantly speeds up the training of graph AE and VAE while preserving or even improving performances. We demonstrate the effectiveness of FastGAE on various real-world graphs, outperforming the few existing approaches to scale graph AE and VAE by a wide marg…
Tsinghua University（清华大学）；UC San Diego（加州大学圣地亚哥分校）；Qualcomm, Inc.（高通公司）；University of Amsterdam（阿姆斯特丹大学）
We propose an algorithm, guided variational autoencoder (Guided-VAE), that is able to learn a controllable generative model by performing latent representation disentanglement learning. The learning objective is achieved by providing signals to the latent encoding/embedding in VAE without changing its main backbone architecture, hence retaining the desirable properties of the VAE. We design an unsupervised strategy and a supervised strategy in Guided-VAE and observe enhanced modeling and controlling capability over the vanilla VAE. In the unsupervised strategy, we guide the VAE learning by int…
DeNA Co., Ltd.（DeNA有限公司）；University of Tokyo（东京大学）；The University of Electro-Communications（电气通信大学）
Variational autoencoder-based voice conversion (VAE-VC) has the advantage of requiring only pairs of speeches and speaker labels for training. Unlike the majority of the research in VAE-VC which focuses on utilizing auxiliary losses or discretizing latent variables, this paper investigates how an increasing model expressiveness has benefits and impacts on the VAE-VC. Specifically, we first analyze VAE-VC from a rate-distortion perspective, and point out that model expressiveness is significant for VAE-VC because rate and distortion reflect similarity and naturalness of converted speeches. Base…
We present fast, realistic image generation on high-resolution, multimodal datasets using hierarchical variational autoencoders (VAEs) trained on a deterministic autoencoder's latent space. In this two-stage setup, the autoencoder compresses the image into its semantic features, which are then modeled with a deep VAE. With this method, the VAE avoids modeling the fine-grained details that constitute the majority of the image's code length, allowing it to focus on learning its structural components. We demonstrate the effectiveness of our two-stage approach, achieving a FID of 9.34 on the Image…
The variational autoencoder (VAE; Kingma, Welling (2014)) is a recently proposed generative model pairing a top-down generative network with a bottom-up recognition network which approximates posterior inference. It typically makes strong assumptions about posterior inference, for instance that the posterior distribution is approximately factorial, and that its parameters can be approximated with nonlinear regression from the observations. As we show empirically, the VAE objective can lead to overly simplified representations which fail to use the network's entire modeling capacity. We present…
Autodesk AI Research（欧特克人工智能研究院）；Imagia（Imagia公司）；MILA（蒙特利尔学习算法研究所）
The latent variables learned by VAEs have seen considerable interest as an unsupervised way of extracting features, which can then be used for downstream tasks. There is a growing interest in the question of whether features learned on one environment will generalize across different environments. We demonstrate here that VAE latent variables often focus on some factors of variation at the expense of others - in this case we refer to the features as ``imbalanced''. Feature imbalance leads to poor generalization when the latent variables are used in an environment where the presence of features…
Variational Autoencoders are one of the most commonly used generative models, particularly for image data. A prominent difficulty in training VAEs is data that is supported on a lower-dimensional manifold. Recent work by Dai and Wipf (2020) proposes a two-stage training algorithm for VAEs, based on a conjecture that in standard VAE training the generator will converge to a solution with 0 variance which is correctly supported on the ground truth manifold. They gave partial support for that conjecture by showing that some optima of the VAE loss do satisfy this property, but did not analyze the …
This research attempts to construct a network that can convert online and offline handwritten characters to each other. The proposed network consists of two Variational Auto-Encoders (VAEs) with a shared latent space. The VAEs are trained to generate online and offline handwritten Latin characters simultaneously. In this way, we create a cross-modal VAE (Cross-VAE). During training, the proposed Cross-VAE is trained to minimize the reconstruction loss of the two modalities, the distribution loss of the two VAEs, and a novel third loss called the space sharing loss. This third, space sharing lo…
Exemplar-based methods rely on informative data points or prototypes to guide the optimization of learning algorithms. Such data facilitate interpretable model design and prediction. Of particular interest is the utility of exemplars in learning unsupervised deep representations. In this paper, we leverage hubs, which emerge as frequent neighbors in high-dimensional spaces, as exemplars to regularize a variational autoencoder and to learn a discriminative embedding for unsupervised down-stream tasks. We propose an unsupervised, data-driven regularization of the latent space with a mixture of h…
Samsung-PDMI Joint AI Center（三星-PDMI联合人工智能中心）；Steklov Institute of Mathematics at St. Petersburg（圣彼得堡斯捷克洛夫数学研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
Recent research has shown the advantages of using autoencoders based on deep neural networks for collaborative filtering. In particular, the recently proposed Mult-VAE model, which used the multinomial likelihood variational autoencoders, has shown excellent results for top-N recommendations. In this work, we propose the Recommender VAE (RecVAE) model that originates from our research on regularization techniques for variational autoencoders. RecVAE introduces several novel ideas to improve Mult-VAE, including a novel composite prior distribution for the latent codes, a new approach to setting…
A new form of the variational autoencoder (VAE) is proposed, based on the symmetric Kullback-Leibler divergence. It is demonstrated that learning of the resulting symmetric VAE (sVAE) has close connections to previously developed adversarial-learning methods. This relationship helps unify the previously distinct techniques of VAE and adversarially learning, and provides insights that allow us to ameliorate shortcomings with some previously developed adversarial methods. In addition to an analysis that motivates and explains the sVAE, an extensive set of experiments validate the utility of the …
Previous work explored blending levels from existing games to create levels for a new game that mixes properties of the original games. In this paper, we use Variational Autoencoders (VAEs) for improving upon such techniques. VAEs are artificial neural networks that learn and use latent representations of datasets to generate novel outputs. We train a VAE on level data from Super Mario Bros. and Kid Icarus, enabling it to capture the latent space spanning both games. We then use this space to generate level segments that combine properties of levels from both games. Moreover, by applying evolu…
Center for the Neural Basis of Cognition, Machine Learning Department（认知神经基础中心，机器学习系）；Carnegie Mellon University（卡内基梅隆大学）
The variational autoencoder (VAE) is a popular probabilistic generative model. However, one shortcoming of VAEs is that the latent variables cannot be discrete, which makes it difficult to generate data from different modes of a distribution. Here, we propose an extension of the VAE framework that incorporates a classifier to infer the discrete class of the modeled data. To model sequential data, we can combine our Classifying VAE with a recurrent neural network such as an LSTM. We apply this model to algorithmic music generation, where our model learns to generate musical sequences in differe…
Deep learning (DL) methods where interpretability is intrinsically considered as part of the model are required to better understand the relationship of clinical and imaging-based attributes with DL outcomes, thus facilitating their use in the reasoning behind medical decisions. Latent space representations built with variational autoencoders (VAE) do not ensure individual control of data attributes. Attribute-based methods enforcing attribute disentanglement have been proposed in the literature for classical computer vision tasks in benchmark data. In this paper, we propose a VAE approach, th…
As a general-purpose generative model architecture, VAE has been widely used in the field of image and natural language processing. VAE maps high dimensional sample data into continuous latent variables with unsupervised learning. Sampling in the latent variable space of the feature, VAE can construct new image or text data. As a general-purpose generation model, the vanilla VAE can not fit well with various data sets and neural networks with different structures. Because of the need to balance the accuracy of reconstruction and the convenience of latent variable sampling in the training proce…
Although variational autoencoders (VAE) are successfully used to obtain meaningful low-dimensional representations for high-dimensional data, the characterization of critical points of the loss function for general observation models is not fully understood. We introduce a theoretical framework that is based on a connection between $β$-VAE and generalized linear models (GLM). The equality between the activation function of a $β$-VAE and the inverse of the link function of a GLM enables us to provide a systematic generalization of the loss analysis for $β$-VAE based on the assumption that the o…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of California, Los Angeles（加州大学洛杉矶分校）
Variational Autoencoders (VAEs) provide a theoretically-backed and popular framework for deep generative models. However, learning a VAE from data poses still unanswered theoretical questions and considerable practical challenges. In this work, we propose an alternative framework for generative modeling that is simpler, easier to train, and deterministic, yet has many of the advantages of VAEs. We observe that sampling a stochastic encoder in a Gaussian VAE can be interpreted as simply injecting noise into the input of a deterministic decoder. We investigate how substituting this kind of stoch…
We extend Stochastic Gradient Variational Bayes to perform posterior inference for the weights of Stick-Breaking processes. This development allows us to define a Stick-Breaking Variational Autoencoder (SB-VAE), a Bayesian nonparametric version of the variational autoencoder that has a latent representation with stochastic dimensionality. We experimentally demonstrate that the SB-VAE, and a semi-supervised variant, learn highly discriminative latent representations that often outperform the Gaussian VAE's.
Recently generative models have focused on combining the advantages of variational autoencoders (VAE) and generative adversarial networks (GAN) for good reconstruction and generative abilities. In this work we introduce a novel hybrid architecture, Implicit Discriminator in Variational Autoencoder (IDVAE), that combines a VAE and a GAN, which does not need an explicit discriminator network. The fundamental premise of the IDVAE architecture is that the encoder of a VAE and the discriminator of a GAN utilize common features and therefore can be trained as a shared network, while the decoder of t…
In today's day and age when almost every industry has an online presence with users interacting in online marketplaces, personalized recommendations have become quite important. Traditionally, the problem of collaborative filtering has been tackled using Matrix Factorization which is linear in nature. We extend the work of [11] on using variational autoencoders (VAEs) for collaborative filtering with implicit feedback by proposing a hybrid, multi-modal approach. Our approach combines movie embeddings (learned from a sibling VAE network) with user ratings from the Movielens 20M dataset and appl…
University of Oxford（牛津大学）；Alan Turing Institute（艾伦·图灵研究所）
We make inroads into understanding the robustness of Variational Autoencoders (VAEs) to adversarial attacks and other input perturbations. While previous work has developed algorithmic approaches to attacking and defending VAEs, there remains a lack of formalization for what it means for a VAE to be robust. To address this, we develop a novel criterion for robustness in probabilistic models: $r$-robustness. We then use this to construct the first theoretical results for the robustness of VAEs, deriving margins in the input space for which we can provide guarantees about the resulting reconstru…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Recent years have witnessed tremendous interest in understanding and predicting information spread on social media platforms such as Twitter, Facebook, etc. Existing diffusion prediction methods primarily exploit the sequential order of influenced users by projecting diffusion cascades onto their local social neighborhoods. However, this fails to capture global social structures that do not explicitly manifest in any of the cascades, resulting in poor performance for inactive users with limited historical activities. In this paper, we present a novel variational autoencoder framework (Inf-VAE)…
