  The problem of symmetric rank-one approximation of symmetric tensors is
important in Independent Components Analysis, also known as Blind Source
Separation, as well as polynomial optimization. We analyze the symmetric
rank-one approximation problem for symmetric tensors and derive several
perturbation results. Given a symmetric rank-one tensor obscured by noise, we
provide bounds on the accuracy of the best symmetric rank-one approximation for
recovering the original rank-one structure, and we show that any eigenvector
with sufficiently large eigenvalue is related to the rank-one structure as
well. Further, we show that for high-dimensional symmetric
approximately-rank-one tensors, the generalized Rayleigh quotient is mostly
close to zero, so the best symmetric rank-one approximation corresponds to a
prominent global extreme value. We show that each iteration of the Shifted
Symmetric Higher Order Power Method (SS-HOPM), when applied to a rank-one
symmetric tensor, moves towards the principal eigenvector for any input and
shift parameter, under mild conditions. Finally, we explore the best choice of
shift parameter for SS-HOPM to recover the principal eigenvector. We show that
SS-HOPM is guaranteed to converge to an eigenvector of an approximately
rank-one even-mode tensor for a wider choice of shift parameter than it is for
a general symmetric tensor. We also show that the principal eigenvector is a
stable fixed point of the SS-HOPM iteration for a wide range of shift
parameters; together with a numerical experiment, these results lead to a
non-obvious recommendation for shift parameter for the symmetric rank-one
approximation problem.
