When optimizing against the mean loss over a distribution of predictions in the context of a regression task, then even if there is a distribution of targets the optimal prediction distribution is always a delta function at a single value. Methods of constructing generative models need to overcome this tendency. We consider a simple method of summarizing the prediction error, such that the optimal strategy corresponds to outputting a distribution of predictions with a support that matches the support of the distribution of targets --- optimizing against the minimal value of the loss given a set of samples from the prediction distribution, rather than the mean. We show that models trained against this loss learn to capture the support of the target distribution and, when combined with an auxiliary classifier-like prediction task, can be projected via rejection sampling to reproduce the full distribution of targets. The resulting method works well compared to other generative modeling approaches particularly in low dimensional spaces with highly non-trivial distributions, due to mode collapse solutions being globally suboptimal with respect to the extreme value loss. However, the method is less suited to high-dimensional spaces such as images due to the scaling of the number of samples needed in order to accurately estimate the extreme value loss when the dimension of the data manifold becomes large.
: One method for generating samples from a high-dimensional distribution is to factorize the joint distribution into a product of one-dimensional conditional probabilities: @math . These one-dimensional distributions can each be learned separately by discretizing the possibilities and minimizing the categorical cross-entropy or KL divergence. Alternately, if the variables @math have some structural relationship, such as words in a sentence or pixels in an image, then weight sharing and causal masking can be used to learn a single model which is able to generate all such distributions. Sampling from the distribution is done by sampling from each one-dimensional distribution in turn, conditioning on the previous samples. Alternately, methods such as beam-search can be used to find maximum likelihood outcomes or otherwise adjust the 'temperature' of the generator. Methods which use this approach include CharRNN @cite_18 , PixelRNN @cite_4 , PixelCNN @cite_25 @cite_28 , WaveNet @cite_3 , and Transformer networks @cite_9 .
@cite_18 @cite_4 @cite_28 @cite_9 @cite_3 @cite_25
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We propose in this paper a new approach to train the Generative Adversarial Nets (GANs) with a mixture of generators to overcome the mode collapsing problem. The main intuition is to employ multiple generators, instead of using a single one as in the original GAN. The idea is simple, yet proven to be extremely effective at covering diverse data modes, easily overcoming the mode collapsing problem and delivering state-of-the-art results. A minimax formulation was able to establish among a classifier, a discriminator, and a set of generators in a similar spirit with GAN. Generators create samples that are intended to come from the same distribution as the training data, whilst the discriminator determines whether samples are true data or generated by generators, and the classifier specifies which generator a sample comes from. The distinguishing feature is that internal samples are created from multiple generators, and then one of them will be randomly selected as final output similar to the mechanism of a probabilistic mixture model. We term our method Mixture Generative Adversarial Nets (MGAN). We develop theoretical analysis to prove that, at the equilibrium, the Jensen-Shannon divergence (JSD) between the mixture of generators’ distributions and the empirical data distribution is minimal, whilst the JSD among generators’ distributions is maximal, hence effectively avoiding the mode collapsing problem. By utilizing parameter sharing, our proposed model adds minimal computational cost to the standard GAN, and thus can also efficiently scale to large-scale datasets. We conduct extensive experiments on synthetic 2D data and natural image databases (CIFAR-10, STL-10 and ImageNet) to demonstrate the superior performance of our MGAN in achieving state-of-the-art Inception scores over latest baselines, generating diverse and appealing recognizable objects at different resolutions, and specializing in capturing different types of objects by the generators. SUMMARY A generalization of the sampling method introduced by (1953) is presented along with an exposition of the relevant theory, techniques of application and methods and difficulties of assessing the error in Monte Carlo estimates. Examples of the methods, including the generation of random orthogonal matrices and potential applications of the methods to numerical problems arising in statistics, are discussed. For numerical problems in a large number of dimensions, Monte Carlo methods are often more efficient than conventional numerical methods. However, implementation of the Monte Carlo methods requires sampling from high dimensional probability distributions and this may be very difficult and expensive in analysis and computer time. General methods for sampling from, or estimating expectations with respect to, such distributions are as follows. (i) If possible, factorize the distribution into the product of one-dimensional conditional distributions from which samples may be obtained. (ii) Use importance sampling, which may also be used for variance reduction. That is, in order to evaluate the integral J = X) p(x)dx = Ev(f), where p(x) is a probability density function, instead of obtaining independent samples XI, ..., Xv from p(x) and using the estimate J, = Zf(xi) N, we instead obtain the sample from a distribution with density q(x) and use the estimate J2 = Y f(xj)p(x1) q(xj)N . This may be advantageous if it is easier to sample from q(x) thanp(x), but it is a difficult method to use in a large number of dimensions, since the values of the weights w(xi) = p(x1) q(xj) for reasonable values of N may all be extremely small, or a few may be extremely large. In estimating the probability of an event A, however, these difficulties may not be as serious since the only values of w(x) which are important are those for which x -A. Since the methods proposed by Trotter & Tukey (1956) for the estimation of conditional expectations require the use of importance sampling, the same difficulties may be encountered in their use. (iii) Use a simulation technique; that is, if it is difficult to sample directly from p(x) or if p(x) is unknown, sample from some distribution q(y) and obtain the sample x values as some function of the corresponding y values. If we want samples from the conditional dis We propose in this paper a novel approach to tackle the problem of mode collapse encountered in generative adversarial network (GAN). Our idea is intuitive but proven to be very effective, especially in addressing some key limitations of GAN. In essence, it combines the Kullback-Leibler (KL) and reverse KL divergences into a unified objective function, thus it exploits the complementary statistical properties from these divergences to effectively diversify the estimated density in capturing multi-modes. We term our method dual discriminator generative adversarial nets (D2GAN) which, unlike GAN, has two discriminators; and together with a generator, it also has the analogy of a minimax game, wherein a discriminator rewards high scores for samples from data distribution whilst another discriminator, conversely, favoring data from the generator, and the generator produces data to fool both two discriminators. We develop theoretical analysis to show that, given the maximal discriminators, optimizing the generator of D2GAN reduces to minimizing both KL and reverse KL divergences between data distribution and the distribution induced from the data generated by the generator, hence effectively avoiding the mode collapsing problem. We conduct extensive experiments on synthetic and real-world large-scale datasets (MNIST, CIFAR-10, STL-10, ImageNet), where we have made our best effort to compare our D2GAN with the latest state-of-the-art GAN's variants in comprehensive qualitative and quantitative evaluations. The experimental results demonstrate the competitive and superior performance of our approach in generating good quality and diverse samples over baselines, and the capability of our method to scale up to ImageNet database. Determining the structure of dependencies among a set of variables is a common task in many signal and image processing applications, including multitarget tracking and computer vision. In this paper, we present an information-theoretic, machine learning approach to problems of this type. We cast this problem as a hypothesis test between factorizations of variables into mutually independent subsets. We show that the likelihood ratio can be written as sums of two sets of Kullback-Leibler (KL) divergence terms. The first set captures the structure of the statistical dependencies within each hypothesis, whereas the second set measures the details of model differences between hypotheses. We then consider the case when the signal prior models are unknown, so that the distributions of interest must be estimated directly from data, showing that the second set of terms is (asymptotically) negligible and quantifying the loss in hypothesis separability when the models are completely unknown. We demonstrate the utility of nonparametric estimation methods for such problems, providing a general framework for determining and distinguishing between dependency structures in highly uncertain environments. Additionally, we develop a machine learning approach for estimating lower bounds on KL divergence and mutual information from samples of high-dimensional random variables for which direct density estimation is infeasible. We present empirical results in the context of three prototypical applications: association of signals generated by sources possessing harmonic behavior, scene correspondence using video imagery, and detection of coherent behavior among sets of moving objects.