Inference and uncertainty quantification for noisy matrix completion
Inference and uncertainty quantification for noisy matrix completion
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DOI：
10.1073/pnas.1910053116
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发表时间：
2019-11-12
影响因子：
11.1
通讯作者：
Yan, Yuling
中科院分区：
文献类型：
--
作者：
Chen, Yuxin;Fan, Jianqing;Yan, Yuling
嘈杂的矩阵完成旨在估计仅给出部分和损坏的条目的低排名矩阵。尽管在设计有效的估计算法方面取得了显着进展，但仍不清楚如何评估所获得的估计值的不确定性以及如何对未知矩阵进行有效的统计推断（例如，构建一个有效且短的置信区间以进行不看者进入）。本文朝着解决此类任务迈出了重大步骤。我们开发了一个简单的程序来补偿广泛使用的凸和非凸估计量的偏差。由此产生的DEBIAS估计器承认几乎精确的非肌分布特征，从而使置信区间/区域的最佳结构能够以缺失的条目和低级别的因素为单位。我们的推论程序不需要样品分割，因此避免了不必要的数据效率丢失。作为副产品，我们获得了率和恒定估计量的估计准确性的尖锐表征。我们的依据估计量是可访问的算法，可证明可以实现完整的统计效率。
Noisy matrix completion aims at estimating a low-rank matrix given only partial and corrupted entries. Despite remarkable progress in designing efficient estimation algorithms, it remains largely unclear how to assess the uncertainty of the obtained estimates and how to perform efficient statistical inference on the unknown matrix (e.g., constructing a valid and short confidence interval for an unseen entry). This paper takes a substantial step toward addressing such tasks. We develop a simple procedure to compensate for the bias of the widely used convex and nonconvex estimators. The resulting debiased estimators admit nearly precise nonasymptotic distributional characterizations, which in turn enable optimal construction of confidence intervals/regions for, say, the missing entries and the low-rank factors. Our inferential procedures do not require sample splitting, thus avoiding unnecessary loss of data efficiency. As a byproduct, we obtain a sharp characterization of the estimation accuracy of our debiased estimators in both rate and constant. Our debiased estimators are tractable algorithms that provably achieve full statistical efficiency.