We show that this correlation exists for practical datasets such as CIFAR-10. Namely, we consider several possible common data augmentations, both spatial and color-based. As predicted by theory, we observe a negative correlation between the Poincaré constant of the (augmented) dataset and the FID score of a GAN trained with this specific data augmentation. Specifically, larger values of the Poincaré constant imply better connectivity and, thus, better GAN quality, resulting in lower FID.
In principle, this could allow us to discover optimal data augmentation protocols beforehand, since evaluating the Poincaré constant is much faster than training a generative model.