Do GANS (Generative Adversarial Nets) actually learn the target distribution? The foundational paper of Goodfellow et al. (2014) suggested they do, if they were given sufficiently large deep nets, sample size, and computation time. A recent theoretical analysis in Arora et al. (2017) raised doubts whether the same holds when discriminator has bounded size. It showed that the training objective can approach its optimum value even if the generated distribution has very low support. In other words, the training objective is unable to prevent mode collapse. The current paper makes two contributions. (1) It proposes a novel test for estimating support size using the birthday paradox of discrete probability. Using this  evidence is presented that well-known GANs approaches do learn distributions of fairly low support.   (2) It theoretically studies encoder-decoder GANs architectures (e.g., BiGAN/ALI), which were proposed to learn more meaningful features via GANs, and consequently to also solve the mode-collapse issue. Our result shows that such encoder-decoder training objectives also cannot guarantee learning of the full distribution because they cannot prevent serious mode collapse. More seriously, they cannot prevent learning meaningless codes for data, contrary to usual intuition.
We propose a support size estimator of GANs's learned distribution to show they indeed suffer from mode collapse, and we prove that encoder-decoder GANs do not avoid the issue as well. The paper attempts to estimate the size of the support for solutions produced by typical GANs experimentally.  This paper proposes a clever new test based on the birthday paradox for measuring diversity in generated sample, with experiment results interpreted to mean that mode collapse is strong in a number of state-of-the-art generative models. The paper uses birthday paradox to show that some GAN architectures generate distributions with fairly low support.