Exact tensor completion with sum-of-squares
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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 R^n from r·Õ(n^1.5) randomly observed entries of the tensor. This bound improves over the previous best one of r·Õ(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.
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