  The approximation of tensors has important applications in various
disciplines, but it remains an extremely challenging task. It is well known
that tensors of higher order can fail to have best low-rank approximations, but
with an important exception that best rank-one approximations always exists.
The most popular approach to low-rank approximation is the alternating least
squares (ALS) method. The convergence of the alternating least squares
algorithm for the rank-one approximation problem is analysed in this paper. In
our analysis we are focusing on the global convergence and the rate of
convergence of the ALS algorithm. It is shown that the ALS method can converge
sublinearly, Q-linearly, and even Q-superlinearly. Our theoretical results are
illustrated on explicit examples.
