Identifiability of Discretized Latent Coordinate Systems via Density Landmarks Detection

Disentanglement aims to recover meaningful latent ground-truth factors from only the observed distribution. Identifiability provides the theoretical grounding for disentanglement to be well-founded. Unfortunately, unsupervised identifiability of independent latent factors is a theoretically proven impossibility in the i.i.d. setting under a general nonlinear smooth map from factors to observations. In this work, we show that, remarkably, it is possible to recover discretized latent coordinates under the most general smooth mapping (diffeomorphism) without any additional inductive bias on the mapping. This is, provided the latent density has axis-aligned discontinuity landmarks, but without making the unrealistic assumption of statistical independence of the factors. We introduce this novel form of identifiability and provide a comprehensive proof of the recovery of discretized coordinates.