Exact nuclear norm, completion and decomposition for random overcomplete tensors via degree-4 SOS

In this paper we show that simple semidefinite programs inspired by degree $4$ SOS can exactly solve the tensor nuclear norm, tensor decomposition, and tensor completion problems on tensors with random asymmetric components. More precisely, for tensor nuclear norm and tensor decomposition, we show that w.h.p. these semidefinite programs can exactly find the nuclear norm and components of an $(n\times n\times n)$-tensor $\mathcal{T}$ with $m\leq n^{3/2}/polylog(n)$ random asymmetric components. For tensor completion, we show that w.h.p. the semidefinite program introduced by Potechin \& Steurer (2017) can exactly recover an $(n\times n\times n)$-tensor $\mathcal{T}$ with $m$ random asymmetric components from only $n^{3/2}m\, polylog(n)$ randomly observed entries. This gives the first theoretical guarantees for exact tensor completion in the overcomplete regime. This matches the best known results for approximate versions of these problems given by Barak \& Moitra (2015) for tensor completion, and Ma, Shi \& Steurer (2016) for tensor decomposition.

Exact tensor completion with sum-of-squares

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.