  This paper studies the rank-$1$ tensor completion problem for cubic order
tensors. First of all, we show that this problem is equivalent to a special
rank-$1$ matrix recovery problem. We propose both nuclear norm relaxation and
moment relaxation methods for solving the resulting rank-$1$ matrix recovery
problem. The nuclear norm relaxation sometimes get a rank-$1$ tensor
completion, while sometimes it does not. When it fails, we apply the moment
hierarchy of semidefinite programming relaxations to solve the rank-$1$ matrix
recovery problem. The moment hierarchy can always get a rank-$1$ tensor
completion, or detect its nonexistence. In particular, when the tensor is
strongly rank-$1$ completable, we show that the problem is equivalent to a
rank-$1$ matrix completion problem and it can be solved by an iterative
formula. Therefore, much larger size problems can be solved efficiently for
strongly rank-$1$ completable tensors. Numerical experiments are shown to
demonstrate the efficiency of these proposed methods.
