When Is Unsupervised Disentanglement Possible?

A common assumption in many domains is that high dimensional data are a smooth nonlinear function of a small number of independent factors. When is it possible to recover the factors from unlabeled data? In the context of deep models this problem is called "disentanglement" and was recently shown to be impossible without additional strong assumptions [17, 19] . In this paper, we show that the assumption of local isometry together with non-Gaussianity of the factors, is sufficient to provably recover disentangled representations from data. We leverage recent advances in deep generative models to construct manifolds of highly realistic images for which the ground truth latent representation is known, and test whether modern and classical methods succeed in recovering the latent factors. For many different manifolds, we find that a spectral method that explicitly optimizes local isometry and non-Gaussianity consistently finds the correct latent factors, while baseline deep autoencoders do not. We propose how to encourage deep autoencoders to find encodings that satisfy local isometry and show that this helps them discover disentangled representations. Overall, our results suggest that in some realistic settings, unsupervised disentanglement is provably possible, without any domain-specific assumptions. 𝑧! 𝑧! 𝑧! 𝑧! 𝑧" 𝑧" 𝑧" 𝑧" 𝑦 ! 𝑦 " 𝑦 ! 𝑦 " (a) cat1 (b) HLLE+ICA (c) IRMAE