  Unsupervised learning of latent variable models (LVMs) is widely used to
represent data in machine learning. When such models reflect the ground truth
factors and the mechanisms mapping them to observations, there is reason to
expect that they allow generalization 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.
