Exact tensor completion with sum-of-squares
with Aaron Potechin. COLT 2017.
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 incoherent, orthogonal components in from randomly observed entries of the tensor. This bound improves over the previous best one of 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 a precisely planted orthogonal global optima over the sphere and that this fact can be certified within the sum-of-squares proof system.
- sum-of-squares method
- tensor computations
- machine learning
- semidefinite programming