Exact tensor completion with sum-of-squares
This provides a more efficient solution for recovering missing data in tensors, which is important for applications like machine learning and data analysis, though it is an incremental advance over matrix completion methods.
The paper tackles the problem of exact tensor completion by developing a polynomial-time algorithm that recovers an unknown 3-tensor with r incoherent, orthogonal components from r·Õ(n^1.5) randomly observed entries, improving over the previous best bound of r·Õ(n^2) and matching results for approximate tensor completion.
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). Our algorithm and analysis extends seminal results for exact matrix completion (Candes & Recht, 2009) to the tensor setting via the sum-of-squares method. The main technical challenge is to show that a small number of randomly chosen monomials are enough to construct a degree-3 polynomial with precisely planted orthogonal global optima over the sphere and that this fact can be certified within the sum-of-squares proof system.