  This work focuses on the problem of unraveling nonlinearly mixed latent
components in an unsupervised manner. The latent components are assumed to
reside in the probability simplex, and are transformed by an unknown
post-nonlinear mixing system. This problem finds various applications in signal
and data analytics, e.g., nonlinear hyperspectral unmixing, image embedding,
and nonlinear clustering. Linear mixture learning problems are already
ill-posed, as identifiability of the target latent components is hard to
establish in general. With unknown nonlinearity involved, the problem is even
more challenging. Prior work offered a function equation-based formulation for
provable latent component identification. However, the identifiability
conditions are somewhat stringent and unrealistic. In addition, the
identifiability analysis is based on the infinite sample (i.e., population)
case, while the understanding for practical finite sample cases has been
elusive. Moreover, the algorithm in the prior work trades model expressiveness
with computational convenience, which often hinders the learning performance.
Our contribution is threefold. First, new identifiability conditions are
derived under largely relaxed assumptions. Second, comprehensive sample
complexity results are presented -- which are the first of the kind. Third, a
constrained autoencoder-based algorithmic framework is proposed for
implementation, which effectively circumvents the challenges in the existing
algorithm. Synthetic and real experiments corroborate our theoretical analyses.
