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Nonlinear ICA is a fundamental problem for unsupervised representationlearning, emphasizing the capacity to recover the underlying latent variablesgenerating the data (i.e., identifiability). Recently, the very firstidentifiability proofs for nonlinear ICA have been proposed, leveraging thetemporal structure of the independent components. Here, we propose a generalframework for nonlinear ICA, which, as a special case, can make use of temporalstructure. It is based on augmenting the data by an auxiliary variable, such asthe time index, the history of the time series, or any other availableinformation. We propose to learn nonlinear ICA by discriminating between trueaugmented data, or data in which the auxiliary variable has been randomized.This enables the framework to be implemented algorithmically through logisticregression, possibly in a neural network. We provide a comprehensive proof ofthe identifiability of the model as well as the consistency of our estimationmethod. The approach not only provides a general theoretical frameworkcombining and generalizing previously proposed nonlinear ICA models andalgorithms, but also brings practical advantages.

Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning