  In this paper, we provide local and global convergence guarantees for
recovering CP (Candecomp/Parafac) tensor decomposition. The main step of the
proposed algorithm is a simple alternating rank-$1$ update which is the
alternating version of the tensor power iteration adapted for asymmetric
tensors. Local convergence guarantees are established for third order tensors
of rank $k$ in $d$ dimensions, when $k=o \bigl( d^{1.5} \bigr)$ and the tensor
components are incoherent. Thus, we can recover overcomplete tensor
decomposition. We also strengthen the results to global convergence guarantees
under stricter rank condition $k \le \beta d$ (for arbitrary constant $\beta >
1$) through a simple initialization procedure where the algorithm is
initialized by top singular vectors of random tensor slices. Furthermore, the
approximate local convergence guarantees for $p$-th order tensors are also
provided under rank condition $k=o \bigl( d^{p/2} \bigr)$. The guarantees also
include tight perturbation analysis given noisy tensor.
