  We study symmetric tensor decompositions, i.e. decompositions of the input
symmetric tensor T of order 3 as sum of r 3rd-order tensor powers of u_i where
u_i are vectors in \C^n. In order to obtain efficient decomposition algorithms,
it is necessary to require additional properties from the u_i. In this paper we
assume that the u_i are linearly independent. This implies that r is at most n,
i.e., the decomposition of T is undercomplete. We will moreover assume that r=n
(we plan to extend this work to the case where r is strictly less than n in a
forthcoming paper). We give a randomized algorithm for the following problem:
given T, an accuracy parameter epsilon, and an upper bound B on the condition
number of the tensor, output vectors u'_i such that u_i and u'_i differ by at
most epsilon (in the l_2 norm and up to permutation and multiplication by
phases) with high probability. The main novel features of our algorithm are:
(1) We provide the first algorithm for this problem that works in the
computation model of finite arithmetic and requires only poly-logarithmic (in
n, B and 1/epsilon) many bits of precision. (2) Moreover, this is also the
first algorithm that runs in linear time in the size of the input tensor. It
requires O(n^3) arithmetic operations for all accuracy parameters epsilon =
1/poly(n).
