  A prominent goal of representation learning research is to achieve
representations which are factorized in a useful manner with respect to the
ground truth factors of variation. The fields of disentangled and equivariant
representation learning have approached this ideal from a range of
complimentary perspectives; however, to date, most approaches have proven to
either be ill-specified or insufficiently flexible to effectively separate all
realistic factors of interest in a learned latent space. In this work, we
propose an alternative viewpoint on such structured representation learning
which we call Flow Factorized Representation Learning, and demonstrate it to
learn both more efficient and more usefully structured representations than
existing frameworks. Specifically, we introduce a generative model which
specifies a distinct set of latent probability paths that define different
input transformations. Each latent flow is generated by the gradient field of a
learned potential following dynamic optimal transport. Our novel setup brings
new understandings to both \textit{disentanglement} and \textit{equivariance}.
We show that our model achieves higher likelihoods on standard representation
learning benchmarks while simultaneously being closer to approximately
equivariant models. Furthermore, we demonstrate that the transformations
learned by our model are flexibly composable and can also extrapolate to new
data, implying a degree of robustness and generalizability approaching the
ultimate goal of usefully factorized representation learning.
