On GANs and GMMs

A longstanding problem in machine learning is to find unsupervised methods that can learn the statistical structure of high dimensional signals. In recent years, GANs have gained much attention as a possible solution to the problem, and in particular have shown the ability to generate remarkably realistic high resolution sampled images. At the same time, many authors have pointed out that GANs may fail to model the full distribution ("mode collapse") and that using the learned models for anything other than generating samples may be very difficult. In this paper, we examine the utility of GANs in learning statistical models of images by comparing them to perhaps the simplest statistical model, the Gaussian Mixture Model. First, we present a simple method to evaluate generative models based on relative proportions of samples that fall into predetermined bins. Unlike previous automatic methods for evaluating models, our method does not rely on an additional neural network nor does it require approximating intractable computations. Second, we compare the performance of GANs to GMMs trained on the same datasets. While GMMs have previously been shown to be successful in modeling small patches of images, we show how to train them on full sized images despite the high dimensionality. Our results show that GMMs can generate realistic samples (although less sharp than those of GANs) but also capture the full distribution, which GANs fail to do. Furthermore, GMMs allow efficient inference and explicit representation of the underlying statistical structure. Finally, we discuss how GMMs can be used to generate sharp images.Comment: Accepted to NIPS 201

Many Paths to Equilibrium: GANs Do Not Need to Decrease a Divergence At Every Step

Generative adversarial networks (GANs) are a family of generative models that do not minimize a single training criterion. Unlike other generative models, the data distribution is learned via a game between a generator (the generative model) and a discriminator (a teacher providing training signal) that each minimize their own cost. GANs are designed to reach a Nash equilibrium at which each player cannot reduce their cost without changing the other players' parameters. One useful approach for the theory of GANs is to show that a divergence between the training distribution and the model distribution obtains its minimum value at equilibrium. Several recent research directions have been motivated by the idea that this divergence is the primary guide for the learning process and that every step of learning should decrease the divergence. We show that this view is overly restrictive. During GAN training, the discriminator provides learning signal in situations where the gradients of the divergences between distributions would not be useful. We provide empirical counterexamples to the view of GAN training as divergence minimization. Specifically, we demonstrate that GANs are able to learn distributions in situations where the divergence minimization point of view predicts they would fail. We also show that gradient penalties motivated from the divergence minimization perspective are equally helpful when applied in other contexts in which the divergence minimization perspective does not predict they would be helpful. This contributes to a growing body of evidence that GAN training may be more usefully viewed as approaching Nash equilibria via trajectories that do not necessarily minimize a specific divergence at each step.Comment: 18 page

The Cramer Distance as a Solution to Biased Wasserstein Gradients

The Wasserstein probability metric has received much attention from the machine learning community. Unlike the Kullback-Leibler divergence, which strictly measures change in probability, the Wasserstein metric reflects the underlying geometry between outcomes. The value of being sensitive to this geometry has been demonstrated, among others, in ordinal regression and generative modelling. In this paper we describe three natural properties of probability divergences that reflect requirements from machine learning: sum invariance, scale sensitivity, and unbiased sample gradients. The Wasserstein metric possesses the first two properties but, unlike the Kullback-Leibler divergence, does not possess the third. We provide empirical evidence suggesting that this is a serious issue in practice. Leveraging insights from probabilistic forecasting we propose an alternative to the Wasserstein metric, the Cram\'er distance. We show that the Cram\'er distance possesses all three desired properties, combining the best of the Wasserstein and Kullback-Leibler divergences. To illustrate the relevance of the Cram\'er distance in practice we design a new algorithm, the Cram\'er Generative Adversarial Network (GAN), and show that it performs significantly better than the related Wasserstein GAN

Is Generator Conditioning Causally Related to GAN Performance?

Recent work (Pennington et al, 2017) suggests that controlling the entire distribution of Jacobian singular values is an important design consideration in deep learning. Motivated by this, we study the distribution of singular values of the Jacobian of the generator in Generative Adversarial Networks (GANs). We find that this Jacobian generally becomes ill-conditioned at the beginning of training. Moreover, we find that the average (with z from p(z)) conditioning of the generator is highly predictive of two other ad-hoc metrics for measuring the 'quality' of trained GANs: the Inception Score and the Frechet Inception Distance (FID). We test the hypothesis that this relationship is causal by proposing a 'regularization' technique (called Jacobian Clamping) that softly penalizes the condition number of the generator Jacobian. Jacobian Clamping improves the mean Inception Score and the mean FID for GANs trained on several datasets. It also greatly reduces inter-run variance of the aforementioned scores, addressing (at least partially) one of the main criticisms of GANs

GILBO: One Metric to Measure Them All

We propose a simple, tractable lower bound on the mutual information contained in the joint generative density of any latent variable generative model: the GILBO (Generative Information Lower BOund). It offers a data-independent measure of the complexity of the learned latent variable description, giving the log of the effective description length. It is well-defined for both VAEs and GANs. We compute the GILBO for 800 GANs and VAEs each trained on four datasets (MNIST, FashionMNIST, CIFAR-10 and CelebA) and discuss the results.Comment: Accepted at NeurIPS 201

On the Discrimination-Generalization Tradeoff in GANs

Generative adversarial training can be generally understood as minimizing certain moment matching loss defined by a set of discriminator functions, typically neural networks. The discriminator set should be large enough to be able to uniquely identify the true distribution (discriminative), and also be small enough to go beyond memorizing samples (generalizable). In this paper, we show that a discriminator set is guaranteed to be discriminative whenever its linear span is dense in the set of bounded continuous functions. This is a very mild condition satisfied even by neural networks with a single neuron. Further, we develop generalization bounds between the learned distribution and true distribution under different evaluation metrics. When evaluated with neural distance, our bounds show that generalization is guaranteed as long as the discriminator set is small enough, regardless of the size of the generator or hypothesis set. When evaluated with KL divergence, our bound provides an explanation on the counter-intuitive behaviors of testing likelihood in GAN training. Our analysis sheds lights on understanding the practical performance of GANs.Comment: ICLR 201

Restricting Greed in Training of Generative Adversarial Network

Generative adversarial network (GAN) has gotten wide re-search interest in the field of deep learning. Variations of GAN have achieved competitive results on specific tasks. However, the stability of training and diversity of generated instances are still worth studying further. Training of GAN can be thought of as a greedy procedure, in which the generative net tries to make the locally optimal choice (minimizing loss function of discriminator) in each iteration. Unfortunately, this often makes generated data resemble only a few modes of real data and rotate between modes. To alleviate these problems, we propose a novel training strategy to restrict greed in training of GAN. With help of our method, the generated samples can cover more instance modes with more stable training process. Evaluating our method on several representative datasets, we demonstrate superiority of improved training strategy on typical GAN models with different distance metrics

Training Generative Reversible Networks

Generative models with an encoding component such as autoencoders currently receive great interest. However, training of autoencoders is typically complicated by the need to train a separate encoder and decoder model that have to be enforced to be reciprocal to each other. To overcome this problem, by-design reversible neural networks (RevNets) had been previously used as generative models either directly optimizing the likelihood of the data under the model or using an adversarial approach on the generated data. Here, we instead investigate their performance using an adversary on the latent space in the adversarial autoencoder framework. We investigate the generative performance of RevNets on the CelebA dataset, showing that generative RevNets can generate coherent faces with similar quality as Variational Autoencoders. This first attempt to use RevNets inside the adversarial autoencoder framework slightly underperformed relative to recent advanced generative models using an autoencoder component on CelebA, but this gap may diminish with further optimization of the training setup of generative RevNets. In addition to the experiments on CelebA, we show a proof-of-principle experiment on the MNIST dataset suggesting that adversary-free trained RevNets can discover meaningful latent dimensions without pre-specifying the number of dimensions of the latent sampling distribution. In summary, this study shows that RevNets can be employed in different generative training settings. Source code for this study is at https://github.com/robintibor/generative-reversibleComment: Source code for this study is at https://github.com/robintibor/generative-reversibl

PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows

As 3D point clouds become the representation of choice for multiple vision and graphics applications, the ability to synthesize or reconstruct high-resolution, high-fidelity point clouds becomes crucial. Despite the recent success of deep learning models in discriminative tasks of point clouds, generating point clouds remains challenging. This paper proposes a principled probabilistic framework to generate 3D point clouds by modeling them as a distribution of distributions. Specifically, we learn a two-level hierarchy of distributions where the first level is the distribution of shapes and the second level is the distribution of points given a shape. This formulation allows us to both sample shapes and sample an arbitrary number of points from a shape. Our generative model, named PointFlow, learns each level of the distribution with a continuous normalizing flow. The invertibility of normalizing flows enables the computation of the likelihood during training and allows us to train our model in the variational inference framework. Empirically, we demonstrate that PointFlow achieves state-of-the-art performance in point cloud generation. We additionally show that our model can faithfully reconstruct point clouds and learn useful representations in an unsupervised manner. The code will be available at https://github.com/stevenygd/PointFlow.Comment: Published in ICCV 201

How Generative Adversarial Networks and Their Variants Work: An Overview

Generative Adversarial Networks (GAN) have received wide attention in the machine learning field for their potential to learn high-dimensional, complex real data distribution. Specifically, they do not rely on any assumptions about the distribution and can generate real-like samples from latent space in a simple manner. This powerful property leads GAN to be applied to various applications such as image synthesis, image attribute editing, image translation, domain adaptation and other academic fields. In this paper, we aim to discuss the details of GAN for those readers who are familiar with, but do not comprehend GAN deeply or who wish to view GAN from various perspectives. In addition, we explain how GAN operates and the fundamental meaning of various objective functions that have been suggested recently. We then focus on how the GAN can be combined with an autoencoder framework. Finally, we enumerate the GAN variants that are applied to various tasks and other fields for those who are interested in exploiting GAN for their research.Comment: 41 pages, 16 figures, Published in ACM Computing Surveys (CSUR