Abstract

We obtain the first polynomial-time algorithm for exact tensor completion that improves over the bound implied by reduction to matrix completion. The algorithm recovers an unknown 3-tensor with $r$ incoherent, orthogonal components in $\mathbb{R}^{n}$ from $r\cdot\tilde{O}(n^{1.5})$ randomly observed entries of the tensor. This bound improves over the previous best one of $r\cdot\tilde{O}(n^{2})$ by reduction to exact matrix completion. Our bound also matches the best known results for the easier problem of approximate tensor completion (Barak & Moitra, 2015).