Low-Rank Tensor Decomposition over Finite Fields

We show that finding rank-$R$ decompositions of a 3D tensor, for $R\le 4$, over a fixed finite field can be done in polynomial time. However, if some cells in the tensor are allowed to have arbitrary values, then rank-2 is NP-hard over the integers modulo 2. We also explore rank-1 decomposition of a 3D tensor and of a matrix where some cells are allowed to have arbitrary values.