STATISTICALLY OPTIMAL AND COMPUTATIONALLY EFFICIENT LOW RANK TENSOR COMPLETION FROM NOISY ENTRIES

• 成果类型: Article
• 署名作者: Xia, Dong; Yuan, Ming; Zhang, Cun-Hui
• 署名单位: Hong Kong University of Science & Technology; Columbia University; Rutgers University System; Rutgers University New Brunswick
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS1942
• 发表日期: 2021
• 页码: 76-99
• 关键词: matrix completion; decompositions; complexity

摘要

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 kth order low rank tensor under the general l(p) (1 <= p <= 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.