  Many idealized problems in signal processing, machine learning and statistics
can be reduced to the problem of finding the symmetric canonical decomposition
of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing
inspiration from the matrix case, the successive rank-one approximations (SROA)
scheme has been proposed and shown to yield this tensor decomposition exactly,
and a plethora of numerical methods have thus been developed for the tensor
rank-one approximation problem. In practice, however, the inevitable errors
(say) from estimation, computation, and modeling necessitate that the input
tensor can only be assumed to be a nearly SOD tensor---i.e., a symmetric tensor
slightly perturbed from the underlying SOD tensor. This article shows that even
in the presence of perturbation, SROA can still robustly recover the symmetric
canonical decomposition of the underlying tensor. It is shown that when the
perturbation error is small enough, the approximation errors do not accumulate
with the iteration number. Numerical results are presented to support the
theoretical findings.
