N ONLINEAR ICA USING V OLUME -P RESERVING T RANSFORMATIONS

Nonlinear ICA is a fundamental problem in machine learning, aiming to identify the underlying independent components (sources) from data which is assumed to be a nonlinear function (mixing function) of these sources. Recent works prove that if the sources have some particular structures (e.g. temporal structure), they are theoretically identifiable even if the mixing function is arbitrary. However, in many cases such restrictions on the sources are difficult to satisfy or even verify, hence it inhibits the applicability of the proposed methods. Different from these works, we propose a general framework for nonlinear ICA, in which the mixing function is assumed to be a volume-preserving transformation, and meanwhile the conditions on the sources can be much looser. We provide an insightful proof of the identifiability of the proposed framework. We implement the framework by volume-preserving flow-based models, and verify our theory by experiments on artificial data and synthesized images. Moreover, results on real-world images indicate that our framework can disentangle interpretable features. and u (2) ; the existence of overlap of samples from p ( s | u (1) ) and p ( s | u (1) ) . For verifying experiment of each synthesize an artificial sources can still be successfully identified. This indicates that there exists stronger identifiability theorem with less conditions, which is an appealing for further exploration. We also empirically show that a volume-preserving flow-based model using two classes significantly excels the state-of-the-art ICA method (iVAE), and is able to learn interpretable attributes from real world datasets. These results show the advantages and applicability of our proposed framework. bright-ness, hair (bang) and grin, respectively. These results further demonstrate that our framework is probably able to identify the true sources from real-world datasets, and hence support our theory.