In this article, we develop methods for estimating a low rank tensor from noisy observations on a subset of its entries to achieve both statistical and computational efficiencies. There have been a lot of recent interests in this problem of noisy tensor completion. Much of the attention has been focused on the fundamental computational challenges often associated with problems involving higher order tensors, yet very little is known about their statistical performance. To fill in this void, in this article, we characterize the fundamental statistical limits of noisy tensor completion by establishing minimax optimal rates of convergence for estimating a $k$th order low rank tensor under the general $\ell _{p}$ ($1\le p\le 2$) norm which suggest significant room for improvement over the existing approaches. Furthermore, we propose a polynomial-time computable estimating procedure based upon power iteration and a second-order spectral initialization that achieves the optimal rates of convergence. Our method is fairly easy to implement and numerical experiments are presented to further demonstrate the practical merits of our estimator.

中文翻译：

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统计上最优且计算效率高的低阶张量补全

在本文中，我们开发了一种方法，可通过对噪声项的子集进行低阶张量估计来估计低秩张量，以实现统计和计算效率。最近，人们对这个张量完成噪声大的问题产​​生了兴趣。大部分注意力都集中在通常与涉及高阶张量的问题相关的基本计算挑战上，但对其统计性能知之甚少。为了填补这一空白，在本文中，我们通过建立最小最大最优收敛率来估计噪声张量完成的基本统计极限，以估计一般$ \ ell _ {p} $（ $ 1 \ le p \ le 2 $）规范，这表明与现有方法相比有很大的改进空间。此外，我们提出了一种基于功率迭代和可实现最优收敛速率的二阶频谱初始化的多项式时间可计算估计程序。我们的方法很容易实现，并且进行了数值实验以进一步证明我们的估计器的实际优点。