On the Identifiability of Nonlinear ICA with Unconditional Priors

Nonlinear independent component analysis (ICA) aims to recover the underlying marginally independent latent sources from their observable nonlinear mixtures. The identifiability of nonlinear ICA is a major unsolved problem in unsupervised learning. Recent breakthroughs reformulate the standard marginal independence assumption of sources as conditional independence given some auxiliary variables (e.g., class labels) as weak supervision or inductive bias. However, the modified setting is not applicable in many scenarios that do not have auxiliary variables. We explore an alternative path and consider instead only assumptions on the mixing process, such as the pairwise orthogonality among the columns of the Jacobian of the mixing function. We show that marginally independent latent sources can be identified from strongly nonlinear mixtures up to a component-wise transformation and a permutation, thus providing, to the best of our knowledge, a first full identifiability result of nonlinear ICA without auxiliary variables. We provide an estimation method and validate the theoretical results experimentally.