Generative Adversarial Nets (GAN) have received considerable attention since the 2014 groundbreaking work by Goodfellow et al. Such attention has led to an explosion in new ideas, techniques and applications of GANs. To better understand GANs we need to understand the mathematical foundation behind them. This paper attempts to provide an overview of GANs from a mathematical point of view. Many students in mathematics may find the papers on GANs more difficulty to fully understand because most of them are written from computer science and engineer point of view. The aim of this paper is to give more mathematically oriented students an introduction to GANs in a language that is more familiar to them.

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Generative Adversarial Networks (GANs) is a novel class of deep generative models which has recently gained significant attention. GANs learns complex and high-dimensional distributions implicitly over images, audio, and data. However, there exists major challenges in training of GANs, i.e., mode collapse, non-convergence and instability, due to inappropriate design of network architecture, use of objective function and selection of optimization algorithm. Recently, to address these challenges, several solutions for better design and optimization of GANs have been investigated based on techniques of re-engineered network architectures, new objective functions and alternative optimization algorithms. To the best of our knowledge, there is no existing survey that has particularly focused on broad and systematic developments of these solutions. In this study, we perform a comprehensive survey of the advancements in GANs design and optimization solutions proposed to handle GANs challenges. We first identify key research issues within each design and optimization technique and then propose a new taxonomy to structure solutions by key research issues. In accordance with the taxonomy, we provide a detailed discussion on different GANs variants proposed within each solution and their relationships. Finally, based on the insights gained, we present the promising research directions in this rapidly growing field.

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Generative adversarial networks (GANs) are a hot research topic recently. GANs have been widely studied since 2014, and a large number of algorithms have been proposed. However, there is few comprehensive study explaining the connections among different GANs variants, and how they have evolved. In this paper, we attempt to provide a review on various GANs methods from the perspectives of algorithms, theory, and applications. Firstly, the motivations, mathematical representations, and structure of most GANs algorithms are introduced in details. Furthermore, GANs have been combined with other machine learning algorithms for specific applications, such as semi-supervised learning, transfer learning, and reinforcement learning. This paper compares the commonalities and differences of these GANs methods. Secondly, theoretical issues related to GANs are investigated. Thirdly, typical applications of GANs in image processing and computer vision, natural language processing, music, speech and audio, medical field, and data science are illustrated. Finally, the future open research problems for GANs are pointed out.

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Inferring the most likely configuration for a subset of variables of a joint distribution given the remaining ones - which we refer to as co-generation - is an important challenge that is computationally demanding for all but the simplest settings. This task has received a considerable amount of attention, particularly for classical ways of modeling distributions like structured prediction. In contrast, almost nothing is known about this task when considering recently proposed techniques for modeling high-dimensional distributions, particularly generative adversarial nets (GANs). Therefore, in this paper, we study the occurring challenges for co-generation with GANs. To address those challenges we develop an annealed importance sampling based Hamiltonian Monte Carlo co-generation algorithm. The presented approach significantly outperforms classical gradient based methods on a synthetic and on the CelebA and LSUN datasets.

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Generative adversarial networks (GANs) have been extensively studied in the past few years. Arguably the revolutionary techniques are in the area of computer vision such as plausible image generation, image to image translation, facial attribute manipulation and similar domains. Despite the significant success achieved in computer vision field, applying GANs over real-world problems still have three main challenges: (1) High quality image generation; (2) Diverse image generation; and (3) Stable training. Considering numerous GAN-related research in the literature, we provide a study on the architecture-variants and loss-variants, which are proposed to handle these three challenges from two perspectives. We propose loss and architecture-variants for classifying most popular GANs, and discuss the potential improvements with focusing on these two aspects. While several reviews for GANs have been presented, there is no work focusing on the review of GAN-variants based on handling challenges mentioned above. In this paper, we review and critically discuss 7 architecture-variant GANs and 9 loss-variant GANs for remedying those three challenges. The objective of this review is to provide an insight on the footprint that current GANs research focuses on the performance improvement. Code related to GAN-variants studied in this work is summarized on https://github.com/sheqi/GAN_Review.

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In this paper we study the convergence of generative adversarial networks (GANs) from the perspective of the informativeness of the gradient of the optimal discriminative function. We show that GANs without restriction on the discriminative function space commonly suffer from the problem that the gradient produced by the discriminator is uninformative to guide the generator. By contrast, Wasserstein GAN (WGAN), where the discriminative function is restricted to $1$-Lipschitz, does not suffer from such a gradient uninformativeness problem. We further show in the paper that the model with a compact dual form of Wasserstein distance, where the Lipschitz condition is relaxed, also suffers from this issue. This implies the importance of Lipschitz condition and motivates us to study the general formulation of GANs with Lipschitz constraint, which leads to a new family of GANs that we call Lipschitz GANs (LGANs). We show that LGANs guarantee the existence and uniqueness of the optimal discriminative function as well as the existence of a unique Nash equilibrium. We prove that LGANs are generally capable of eliminating the gradient uninformativeness problem. According to our empirical analysis, LGANs are more stable and generate consistently higher quality samples compared with WGAN.

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Generative Adversarial networks (GANs) have obtained remarkable success in many unsupervised learning tasks and unarguably, clustering is an important unsupervised learning problem. While one can potentially exploit the latent-space back-projection in GANs to cluster, we demonstrate that the cluster structure is not retained in the GAN latent space. In this paper, we propose ClusterGAN as a new mechanism for clustering using GANs. By sampling latent variables from a mixture of one-hot encoded variables and continuous latent variables, coupled with an inverse network (which projects the data to the latent space) trained jointly with a clustering specific loss, we are able to achieve clustering in the latent space. Our results show a remarkable phenomenon that GANs can preserve latent space interpolation across categories, even though the discriminator is never exposed to such vectors. We compare our results with various clustering baselines and demonstrate superior performance on both synthetic and real datasets.

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Recently introduced generative adversarial network (GAN) has been shown numerous promising results to generate realistic samples. The essential task of GAN is to control the features of samples generated from a random distribution. While the current GAN structures, such as conditional GAN, successfully generate samples with desired major features, they often fail to produce detailed features that bring specific differences among samples. To overcome this limitation, here we propose a controllable GAN (ControlGAN) structure. By separating a feature classifier from a discriminator, the generator of ControlGAN is designed to learn generating synthetic samples with the specific detailed features. Evaluated with multiple image datasets, ControlGAN shows a power to generate improved samples with well-controlled features. Furthermore, we demonstrate that ControlGAN can generate intermediate features and opposite features for interpolated and extrapolated input labels that are not used in the training process. It implies that ControlGAN can significantly contribute to the variety of generated samples.

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Quantum machine learning is expected to be one of the first potential general-purpose applications of near-term quantum devices. A major recent breakthrough in classical machine learning is the notion of generative adversarial training, where the gradients of a discriminator model are used to train a separate generative model. In this work and a companion paper, we extend adversarial training to the quantum domain and show how to construct generative adversarial networks using quantum circuits. Furthermore, we also show how to compute gradients -- a key element in generative adversarial network training -- using another quantum circuit. We give an example of a simple practical circuit ansatz to parametrize quantum machine learning models and perform a simple numerical experiment to demonstrate that quantum generative adversarial networks can be trained successfully.

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We introduce an effective model to overcome the problem of mode collapse when training Generative Adversarial Networks (GAN). Firstly, we propose a new generator objective that finds it better to tackle mode collapse. And, we apply an independent Autoencoders (AE) to constrain the generator and consider its reconstructed samples as "real" samples to slow down the convergence of discriminator that enables to reduce the gradient vanishing problem and stabilize the model. Secondly, from mappings between latent and data spaces provided by AE, we further regularize AE by the relative distance between the latent and data samples to explicitly prevent the generator falling into mode collapse setting. This idea comes when we find a new way to visualize the mode collapse on MNIST dataset. To the best of our knowledge, our method is the first to propose and apply successfully the relative distance of latent and data samples for stabilizing GAN. Thirdly, our proposed model, namely Generative Adversarial Autoencoder Networks (GAAN), is stable and has suffered from neither gradient vanishing nor mode collapse issues, as empirically demonstrated on synthetic, MNIST, MNIST-1K, CelebA and CIFAR-10 datasets. Experimental results show that our method can approximate well multi-modal distribution and achieve better results than state-of-the-art methods on these benchmark datasets. Our model implementation is published here: https://github.com/tntrung/gaan

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Deep neural networks (DNNs) have been found to be vulnerable to adversarial examples resulting from adding small-magnitude perturbations to inputs. Such adversarial examples can mislead DNNs to produce adversary-selected results. Different attack strategies have been proposed to generate adversarial examples, but how to produce them with high perceptual quality and more efficiently requires more research efforts. In this paper, we propose AdvGAN to generate adversarial examples with generative adversarial networks (GANs), which can learn and approximate the distribution of original instances. For AdvGAN, once the generator is trained, it can generate adversarial perturbations efficiently for any instance, so as to potentially accelerate adversarial training as defenses. We apply AdvGAN in both semi-whitebox and black-box attack settings. In semi-whitebox attacks, there is no need to access the original target model after the generator is trained, in contrast to traditional white-box attacks. In black-box attacks, we dynamically train a distilled model for the black-box model and optimize the generator accordingly. Adversarial examples generated by AdvGAN on different target models have high attack success rate under state-of-the-art defenses compared to other attacks. Our attack has placed the first with 92.76% accuracy on a public MNIST black-box attack challenge.

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