Push-forwards of Gaussian or log-concave latent variables through Lipschitz neural networks are always sub-Gaussian or sub-exponential, so common deep generative models cannot generate heavy-tailed distributions.
On some theoretical limitations of Generative Adversarial Networks
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abstract
Generative Adversarial Networks have become a core technique in Machine Learning to generate unknown distributions from data samples. They have been used in a wide range of context without paying much attention to the possible theoretical limitations of those models. Indeed, because of the universal approximation properties of Neural Networks, it is a general assumption that GANs can generate any probability distribution. Recently, people began to question this assumption and this article is in line with this thinking. We provide a new result based on Extreme Value Theory showing that GANs can't generate heavy tailed distributions. The full proof of this result is given.
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On the Statistical Capacity of Deep Generative Models
Push-forwards of Gaussian or log-concave latent variables through Lipschitz neural networks are always sub-Gaussian or sub-exponential, so common deep generative models cannot generate heavy-tailed distributions.