Nonlinear independent component analysis (ICA) aims to uncover the true latent sources from their observable nonlinear mixtures. Despite its significance, the identifiability of nonlinear ICA is known to be impossible without additional assumptions. Recent advances have proposed conditions on the connective structure from sources to observed variables, known as Structural Sparsity, to achieve identifiability in an unsupervised manner. However, the sparsity constraint may not hold universally for all sources in practice. Furthermore, the assumptions of bijectivity of the mixing process and independence among all sources, which arise from the setting of ICA, may also be violated in many real-world scenarios. To address these limitations and generalize nonlinear ICA, we propose a set of new identifiability results in the general settings of undercompleteness, partial sparsity and source dependence, and flexible grouping structures. Specifically, we prove identifiability when there are more observed variables than sources (undercomplete), and when certain sparsity and/or source independence assumptions are not met for some changing sources. Moreover, we show that even in cases with flexible grouping structures (e.g., part of the sources can be divided into irreducible independent groups with various sizes), appropriate identifiability results can also be established. Theoretical claims are supported empirically on both synthetic and real-world datasets.

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