Solving exact and noisy rank-one tensor completion with semidefinite programming

Consider recovering a rank-one tensor of size $n_1 \times \cdots \times n_d$ from exact or noisy observations of a few of its entries. We tackle this problem via semidefinite programming (SDP). We derive deterministic combinatorial conditions on the observation mask $\Omega$ (the set of observed indices) under which our SDPs solve the exact completion and achieve robust recovery in the noisy regime. These conditions can be met with as few as $\bigl(\sum_{i=1}^d n_i\bigr) - d + 1$ observations for special $\Omega$. When $\Omega$ is uniformly random, our conditions hold with $O\!\bigl((\prod_{i=1}^d n_i)^{1/2}\,\mathrm{polylog}(\prod_{i=1}^d n_i)\bigr)$ observations. Prior works mostly focus on the uniformly random case, ignoring the practical relevance of structured masks. For $d=2$ (matrix completion), our propagation condition holds if and only if the completion problem admits a unique solution. Our results apply to tensors of arbitrary order and cover both exact and noisy settings. In contrast to much of the literature, our guarantees rely solely on the combinatorial structure of the observation mask, without incoherence assumptions on the ground-truth tensor or uniform randomness of the samples. Preliminary computational experiments show that our SDP methods solve tensor completion problems using significantly fewer observations than alternative methods.