  In this work, we propose a novel method for tensorial independent component
analysis. Our approach is based on TJADE and $ k $-JADE, two recently proposed
generalizations of the classical JADE algorithm. Our novel method achieves the
consistency and the limiting distribution of TJADE under mild assumptions, and
at the same time offers notable improvement in computational speed. Detailed
mathematical proofs of the statistical properties of our method are given and,
as a special case, a conjecture on the properties of $ k $-JADE is resolved.
Simulations and timing comparisons demonstrate remarkable gain in speed.
Moreover, the desired efficiency is obtained approximately for finite samples.
The method is applied successfully to large-scale video data, for which neither
TJADE nor $ k $-JADE is feasible. Finally, an experimental procedure is
proposed to select the values of a set of tuning parameters.
