  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.
