Some Theoretical Properties of GANs

Generative Adversarial Networks (GANs) are a class of generative algorithms that have been shown to produce state-of-the art samples, especially in the domain of image creation. The fundamental principle of GANs is to approximate the unknown distribution of a given data set by optimizing an objective function through an adversarial game between a family of generators and a family of discriminators. In this paper, we offer a better theoretical understanding of GANs by analyzing some of their mathematical and statistical properties. We study the deep connection between the adversarial principle underlying GANs and the Jensen-Shannon divergence, together with some optimality characteristics of the problem. An analysis of the role of the discriminator family via approximation arguments is also provided. In addition, taking a statistical point of view, we study the large sample properties of the estimated distribution and prove in particular a central limit theorem. Some of our results are illustrated with simulated examples

Some Theoretical Insights into Wasserstein GANs

Generative Adversarial Networks (GANs) have been successful in producing outstanding results in areas as diverse as image, video, and text generation. Building on these successes, a large number of empirical studies have validated the benefits of the cousin approach called Wasserstein GANs (WGANs), which brings stabilization in the training process. In the present paper, we add a new stone to the edifice by proposing some theoretical advances in the properties of WGANs. First, we properly define the architecture of WGANs in the context of integral probability metrics parameterized by neural networks and highlight some of their basic mathematical features. We stress in particular interesting optimization properties arising from the use of a parametric 1-Lipschitz discriminator. Then, in a statistically-driven approach, we study the convergence of empirical WGANs as the sample size tends to infinity, and clarify the adversarial effects of the generator and the discriminator by underlining some trade-off properties. These features are finally illustrated with experiments using both synthetic and real-world datasets

GANs with Conditional Independence Graphs: On Subadditivity of Probability Divergences

Generative Adversarial Networks (GANs) are modern methods to learn the underlying distribution of a data set. GANs have been widely used in sample synthesis, de-noising, domain transfer, etc. GANs, however, are designed in a model-free fashion where no additional information about the underlying distribution is available. In many applications, however, practitioners have access to the underlying independence graph of the variables, either as a Bayesian network or a Markov Random Field (MRF). We ask: how can one use this additional information in designing model-based GANs? In this paper, we provide theoretical foundations to answer this question by studying subadditivity properties of probability divergences, which establish upper bounds on the distance between two high-dimensional distributions by the sum of distances between their marginals over (local) neighborhoods of the graphical structure of the Bayes-net or the MRF. We prove that several popular probability divergences satisfy some notion of subadditivity under mild conditions. These results lead to a principled design of a model-based GAN that uses a set of simple discriminators on the neighborhoods of the Bayes-net/MRF, rather than a giant discriminator on the entire network, providing significant statistical and computational benefits. Our experiments on synthetic and real-world datasets demonstrate the benefits of our principled design of model-based GANs

Generative Adversarial Networks (GANs): What it can generate and What it cannot?

In recent years, Generative Adversarial Networks (GANs) have received significant attention from the research community. With a straightforward implementation and outstanding results, GANs have been used for numerous applications. Despite the success, GANs lack a proper theoretical explanation. These models suffer from issues like mode collapse, non-convergence, and instability during training. To address these issues, researchers have proposed theoretically rigorous frameworks inspired by varied fields of Game theory, Statistical theory, Dynamical systems, etc. In this paper, we propose to give an appropriate structure to study these contributions systematically. We essentially categorize the papers based on the issues they raise and the kind of novelty they introduce to address them. Besides, we provide insight into how each of the discussed articles solves the concerned problems. We compare and contrast different results and put forth a summary of theoretical contributions about GANs with focus on image/visual applications. We expect this summary paper to give a bird's eye view to a person wishing to understand the theoretical progress in GANs so far

Relaxed Wasserstein with Applications to GANs

Wasserstein Generative Adversarial Networks (WGANs) provide a versatile class of models, which have attracted great attention in various applications. However, this framework has two main drawbacks: (i) Wasserstein-1 (or Earth-Mover) distance is restrictive such that WGANs cannot always fit data geometry well; (ii) It is difficult to achieve fast training of WGANs. In this paper, we propose a new class of \textit{Relaxed Wasserstein} (RW) distances by generalizing Wasserstein-1 distance with Bregman cost functions. We show that RW distances achieve nice statistical properties while not sacrificing the computational tractability. Combined with the GANs framework, we develop Relaxed WGANs (RWGANs) which are not only statistically flexible but can be approximated efficiently using heuristic approaches. Experiments on real images demonstrate that the RWGAN with Kullback-Leibler (KL) cost function outperforms other competing approaches, e.g., WGANs, even with gradient penalty.Comment: Accepted by ICASSP 2021; add the reference

Non-parametric estimation of Jensen-Shannon Divergence in Generative Adversarial Network training

Generative Adversarial Networks (GANs) have become a widely popular framework for generative modelling of high-dimensional datasets. However their training is well-known to be difficult. This work presents a rigorous statistical analysis of GANs providing straight-forward explanations for common training pathologies such as vanishing gradients. Furthermore, it proposes a new training objective, Kernel GANs, and demonstrates its practical effectiveness on large-scale real-world data sets. A key element in the analysis is the distinction between training with respect to the (unknown) data distribution, and its empirical counterpart. To overcome issues in GAN training, we pursue the idea of smoothing the Jensen-Shannon Divergence (JSD) by incorporating noise in the input distributions of the discriminator. As we show, this effectively leads to an empirical version of the JSD in which the true and the generator densities are replaced by kernel density estimates, which leads to Kernel GANs

The Numerics of GANs

In this paper, we analyze the numerics of common algorithms for training Generative Adversarial Networks (GANs). Using the formalism of smooth two-player games we analyze the associated gradient vector field of GAN training objectives. Our findings suggest that the convergence of current algorithms suffers due to two factors: i) presence of eigenvalues of the Jacobian of the gradient vector field with zero real-part, and ii) eigenvalues with big imaginary part. Using these findings, we design a new algorithm that overcomes some of these limitations and has better convergence properties. Experimentally, we demonstrate its superiority on training common GAN architectures and show convergence on GAN architectures that are known to be notoriously hard to train

The Inductive Bias of Restricted f-GANs

Generative adversarial networks are a novel method for statistical inference that have achieved much empirical success; however, the factors contributing to this success remain ill-understood. In this work, we attempt to analyze generative adversarial learning -- that is, statistical inference as the result of a game between a generator and a discriminator -- with the view of understanding how it differs from classical statistical inference solutions such as maximum likelihood inference and the method of moments. Specifically, we provide a theoretical characterization of the distribution inferred by a simple form of generative adversarial learning called restricted f-GANs -- where the discriminator is a function in a given function class, the distribution induced by the generator is restricted to lie in a pre-specified distribution class and the objective is similar to a variational form of the f-divergence. A consequence of our result is that for linear KL-GANs -- that is, when the discriminator is a linear function over some feature space and f corresponds to the KL-divergence -- the distribution induced by the optimal generator is neither the maximum likelihood nor the method of moments solution, but an interesting combination of both

PacGAN: The power of two samples in generative adversarial networks

Generative adversarial networks (GANs) are innovative techniques for learning generative models of complex data distributions from samples. Despite remarkable recent improvements in generating realistic images, one of their major shortcomings is the fact that in practice, they tend to produce samples with little diversity, even when trained on diverse datasets. This phenomenon, known as mode collapse, has been the main focus of several recent advances in GANs. Yet there is little understanding of why mode collapse happens and why existing approaches are able to mitigate mode collapse. We propose a principled approach to handling mode collapse, which we call packing. The main idea is to modify the discriminator to make decisions based on multiple samples from the same class, either real or artificially generated. We borrow analysis tools from binary hypothesis testing---in particular the seminal result of Blackwell [Bla53]---to prove a fundamental connection between packing and mode collapse. We show that packing naturally penalizes generators with mode collapse, thereby favoring generator distributions with less mode collapse during the training process. Numerical experiments on benchmark datasets suggests that packing provides significant improvements in practice as well.Comment: 49 pages, 24 figure

Implicit Manifold Learning on Generative Adversarial Networks

This paper raises an implicit manifold learning perspective in Generative Adversarial Networks (GANs), by studying how the support of the learned distribution, modelled as a submanifold $\mathcal{M}_{\theta}$, perfectly match with $\mathcal{M}_{r}$, the support of the real data distribution. We show that optimizing Jensen-Shannon divergence forces $\mathcal{M}_{\theta}$ to perfectly match with $\mathcal{M}_{r}$, while optimizing Wasserstein distance does not. On the other hand, by comparing the gradients of the Jensen-Shannon divergence and the Wasserstein distances ($W_1$ and $W_2^2$) in their primal forms, we conjecture that Wasserstein $W_2^2$ may enjoy desirable properties such as reduced mode collapse. It is therefore interesting to design new distances that inherit the best from both distances