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.
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