  We study the classical problem of recovering a multidimensional source signal
from observations of nonlinear mixtures of this signal. We show that this
recovery is possible (up to a permutation and monotone scaling of the source's
original component signals) if the mixture is due to a sufficiently
differentiable and invertible but otherwise arbitrarily nonlinear function and
the component signals of the source are statistically independent with
'non-degenerate' second-order statistics. The latter assumption requires the
source signal to meet one of three regularity conditions which essentially
ensure that the source is sufficiently far away from the non-recoverable
extremes of being deterministic or constant in time. These assumptions, which
cover many popular time series models and stochastic processes, allow us to
reformulate the initial problem of nonlinear blind source separation as a
simple-to-state problem of optimisation-based function approximation. We
propose to solve this approximation problem by minimizing a novel type of
objective function that efficiently quantifies the mutual statistical
dependence between multiple stochastic processes via cumulant-like statistics.
This yields a scalable and direct new method for nonlinear Independent
Component Analysis with widely applicable theoretical guarantees and for which
our experiments indicate good performance.
