  Independent component analysis (ICA) has been a popular dimension reduction
tool in statistical machine learning and signal processing. In this paper, we
present a convergence analysis for an online tensorial ICA algorithm, by
viewing the problem as a nonconvex stochastic approximation problem. For
estimating one component, we provide a dynamics-based analysis to prove that
our online tensorial ICA algorithm with a specific choice of stepsize achieves
a sharp finite-sample error bound. In particular, under a mild assumption on
the data-generating distribution and a scaling condition such that $d^4 / T$ is
sufficiently small up to a polylogarithmic factor of data dimension $d$ and
sample size $T$, a sharp finite-sample error bound of $\tilde O(\sqrt{d / T})$
can be obtained. As a by-product, we also design an online tensorial ICA
algorithm that estimates multiple independent components in parallel, achieving
desirable finite-sample error bound for each independent component estimator.
