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# Nonlinear ICA of Temporally Dependent Stationary Sources

*Proceedings of the 20th International Conference on Artificial Intelligence and Statistics*, PMLR 54:460-469, 2017.

#### Abstract

We develop a nonlinear generalization of independent component analysis (ICA) or blind source separation, based on temporal dependencies (e.g. autocorrelations). We introduce a nonlinear generative model where the independent sources are assumed to be temporally dependent, non-Gaussian, and stationary, and we observe arbitrarily nonlinear mixtures of them. We develop a method for estimating the model (i.e. separating the sources) based on logistic regression in a neural network which learns to discriminate between a short temporal window of the data vs. a temporal window of temporally permuted data. We prove that the method estimates the sources for general smooth mixing nonlinearities, assuming the sources have sufficiently strong temporal dependencies, and these dependencies are in a certain way different from dependencies found in Gaussian processes. For Gaussian (and similar) sources, the method estimates the nonlinear part of the mixing. We thus provide the first rigorous and general proof of identifiability of nonlinear ICA for temporally dependent sources, together with a practical method for its estimation.