论文标题
非凸张量完成的不确定性定量：置信区间，异方差和最佳性
Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality
论文作者
论文摘要
我们研究了噪声张量完成的非凸优化的分布和不确定性 - 鉴于其条目的不完整和损坏的观察结果，估计低量张量的问题。专注于Cai等人提出的两阶段估计算法。 （2019年），我们表征了该非凸估计器的分布，以下到细大尺度。这种分布理论反过来允许人们为看不见的张量条目和未知张量因子构建有效和短的置信区间。提出的推论过程享有几个重要特征：（1）它完全适应噪声异方差，并且（2）它是数据驱动的，并且自动适应未知的噪声分布。此外，我们的发现揭示了非凸张张量完成的统计最佳性：它在估计未知张量和基础张量因子时，达到了不可分解的$ \ ell_ {2} $精度 - 包括速率和预构剂。
We study the distribution and uncertainty of nonconvex optimization for noisy tensor completion -- the problem of estimating a low-rank tensor given incomplete and corrupted observations of its entries. Focusing on a two-stage estimation algorithm proposed by Cai et al. (2019), we characterize the distribution of this nonconvex estimator down to fine scales. This distributional theory in turn allows one to construct valid and short confidence intervals for both the unseen tensor entries and the unknown tensor factors. The proposed inferential procedure enjoys several important features: (1) it is fully adaptive to noise heteroscedasticity, and (2) it is data-driven and automatically adapts to unknown noise distributions. Furthermore, our findings unveil the statistical optimality of nonconvex tensor completion: it attains un-improvable $\ell_{2}$ accuracy -- including both the rates and the pre-constants -- when estimating both the unknown tensor and the underlying tensor factors.
