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.

中文翻译：

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通过 4 阶 SOS 实现随机过完备张量的精确核范数、完成和分解

在本文中，我们展示了受度 $4$ SOS 启发的简单半定程序可以精确地解决具有随机非对称分量的张量上的张量核范数、张量分解和张量完成问题。更准确地说，对于张量核范数和张量分解，我们表明 whp 这些半定程序可以准确地找到 $(n\times n\times n)$-张量 $\mathcal{T}$ 的核范数和分量m\leq n^{3/2}/polylog(n)$ 随机不对称分量。对于张量补全，我们证明了 Potechin \& Steurer (2017) 引入的半定规划可以精确地恢复 $(n\times n\times n)$-张量 $\mathcal{T}$ 与 $m$ 随机不对称来自仅 $n^{3/2}m\、polylog(n)$ 随机观察条目的组件。这为在过完备机制中精确完成张量提供了第一个理论保证。这与 Barak \& Moitra (2015) 给出的张量完成和 Ma, Shi \& Steurer (2016) 给出的张量分解的近似版本的最佳已知结果相匹配。