FONDUE: an algorithm to find the optimal dimensionality of the latent representations of variational autoencoders

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. "How many latent variables should I use for this model?" is a question that many practitioners using variational autoencoders (VAEs) or autoencoders (AEs) have to deal with. When the task has been studied before, this information is available in the literature for the specific architecture and dataset used. However, when it has not, answering this question becomes more complicated.