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.