Function Classes for Identifiable Nonlinear Independent Component Analysis

Simon Buchholz, Michel Besserve, Bernhard Schölkopf

Keywords: nonlinear ICA, independent component analysis, identifiability, blind source separation, unsupervised representation learning, conformal maps

TL;DR: We prove identifiability results for nonlinear Independent Component Analysis in constrained function classes

Abstract: Unsupervised learning of latent variable models (LVMs) is widely used to represent data in machine learning. When such model reflects the ground truth factors and the mechanisms mapping them to observations, there is reason to expect that such models allow generalisation in downstream tasks. It is however well known that such identifiability guaranties are typically not achievable without putting constraints on the model class. This is notably the case for nonlinear Independent Component Analysis, in which the LVM maps statistically independent variables to observations via a deterministic nonlinear function. Several families of spurious solutions fitting perfectly the data, but that do not correspond to the ground truth factors can be constructed in generic settings. However, recent work suggests that constraining the function class of such models may promote identifiability. Specifically, function classes with constraints on their partial derivatives, gathered in the Jacobian matrix, have been proposed, such as orthogonal coordinate transformations (OCT), which impose orthogonality of the Jacobian columns. In the present work, we prove that a subclass of these transformations, conformal maps, is identifiable and provide novel theoretical results suggesting that OCTs have properties that prevent families of spurious solutions to spoil identifiability in a generic setting.