  Disentangled representation learning is one of the major goals of deep
learning, and is a key step for achieving explainable and generalizable models.
A well-defined theoretical guarantee still lacks for the VAE-based unsupervised
methods, which are a set of popular methods to achieve unsupervised
disentanglement. The Group Theory based definition of representation
disentanglement mathematically connects the data transformations to the
representations using the formalism of group. In this paper, built on the
group-based definition and inspired by the n-th dihedral group, we first
propose a theoretical framework towards achieving unsupervised representation
disentanglement. We then propose a model, based on existing VAE-based methods,
to tackle the unsupervised learning problem of the framework. In the
theoretical framework, we prove three sufficient conditions on model, group
structure, and data respectively in an effort to achieve, in an unsupervised
way, disentangled representation per group-based definition. With the first two
of the conditions satisfied and a necessary condition derived for the third
one, we offer additional constraints, from the perspective of the group-based
definition, for the existing VAE-based models. Experimentally, we train 1800
models covering the most prominent VAE-based methods on five datasets to verify
the effectiveness of our theoretical framework. Compared to the original
VAE-based methods, these Groupified VAEs consistently achieve better mean
performance with smaller variances.
