# Exact tensor completion with sum-of-squares

with Aaron Potechin. **COLT 2017.**

## 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 $\R^n$ 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 a precisely planted orthogonal global optima over the sphere and that this fact can be certified within the sum-of-squares proof system.

## keywords

- sum-of-squares method
- tensor computations
- eigenvalues
- machine learning
- semidefinite programming