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. Unlike most of the previous algorithms, our algorithm provides a certificate for the decomposition, does not require knowledge about the number of components in the decomposition and does not make any assumptions on the sizes of the coefficients in the decomposition. As a byproduct, we show that w.h.p. the nuclear norm decomposition exactly coincides with the minimum rank decomposition for tensors 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) for tensors with orthogonal components, 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. For non-orthogonal tensors, this improves the dependence on $m$ of the number of entries needed for exact recovery over all previously known algorithms and provides the first theoretical guarantees for exact tensor completion in the overcomplete regime.