  The key idea of the state-of-the-art VAE-based unsupervised representation
disentanglement methods is to minimize the total correlation of the latent
variable distributions. However, it has been proved that VAE-based unsupervised
disentanglement can not be achieved without introducing other inductive bias.
In this paper, we address VAE-based unsupervised disentanglement by leveraging
the constraints derived from the Group Theory based definition as the
non-probabilistic inductive bias. More specifically, inspired by the nth
dihedral group (the permutation group for regular polygons), we propose a
specific form of the definition and prove its two equivalent conditions:
isomorphism and "the constancy of permutations". We further provide an
implementation of isomorphism based on two Group constraints: the Abel
constraint for the exchangeability and Order constraint for the cyclicity. We
then convert them into a self-supervised training loss that can be incorporated
into VAE-based models to bridge their gaps from the Group Theory based
definition. We train 1800 models covering the most prominent VAE-based models
on five datasets to verify the effectiveness of our method. Compared to the
original models, the Groupidied VAEs consistently achieve better mean
performance with smaller variances, and make meaningful dimensions
controllable.
