  This work introduces a novel principle for disentanglement we call mechanism
sparsity regularization, which applies when the latent factors of interest
depend sparsely on observed auxiliary variables and/or past latent factors. We
propose a representation learning method that induces disentanglement by
simultaneously learning the latent factors and the sparse causal graphical
model that explains them. We develop a nonparametric identifiability theory
that formalizes this principle and shows that the latent factors can be
recovered by regularizing the learned causal graph to be sparse. More
precisely, we show identifiablity up to a novel equivalence relation we call
"consistency", which allows some latent factors to remain entangled (hence the
term partial disentanglement). To describe the structure of this entanglement,
we introduce the notions of entanglement graphs and graph preserving functions.
We further provide a graphical criterion which guarantees complete
disentanglement, that is identifiability up to permutations and element-wise
transformations. We demonstrate the scope of the mechanism sparsity principle
as well as the assumptions it relies on with several worked out examples. For
instance, the framework shows how one can leverage multi-node interventions
with unknown targets on the latent factors to disentangle them. We further draw
connections between our nonparametric results and the now popular exponential
family assumption. Lastly, we propose an estimation procedure based on
variational autoencoders and a sparsity constraint and demonstrate it on
various synthetic datasets. This work is meant to be a significantly extended
version of Lachapelle et al. (2022).
