A trivial example of an equivalence relation is equality ( =). More useful examples in the context of ICA are equivalence up to permutation, rescaling, or scalar transformation. Defining an appropriate equivalence class for the problem at hand therefore allows us to specify exactly the type of indeterminancies which cannot be resolved and up to which the true generative process can be recovered.

Independent component analysis provides a principled framework for unsupervised representation learning, with solid theory on the identifiability of the latent code that generated the data, given only observations of mixtures thereof. Unfortunately, when the mixing is nonlinear, the model is provably nonidentifiable, since statistical independence alone does not sufficiently constrain the problem. Identifiability can be recovered in settings where additional, typically observed variables are included in the generative process. We investigate an alternative path and consider instead including assumptions reflecting the principle of independent causal mechanisms exploited in the field of causality. Specifically, our approach is motivated by thinking of each source as independently influencing the mixing process. This gives rise to a framework which we term independent mechanism analysis. We provide theoretical and empirical evidence that our approach circumvents a number of nonidentifiability issues arising in nonlinear blind source separation.