  When training a variational autoencoder (VAE) on a given dataset, determining
the optimal number of latent variables is mostly done by grid search: a costly
process in terms of computational time and carbon footprint. In this paper, we
explore the intrinsic dimension estimation (IDE) of the data and latent
representations learned by VAEs. We show that the discrepancies between the IDE
of the mean and sampled representations of a VAE after only a few steps of
training reveal the presence of passive variables in the latent space, which,
in well-behaved VAEs, indicates a superfluous number of dimensions. Using this
property, we propose FONDUE: an algorithm which quickly finds the number of
latent dimensions after which the mean and sampled representations start to
diverge (i.e., when passive variables are introduced), providing a principled
method for selecting the number of latent dimensions for VAEs and autoencoders.
