Uncertainty Quantification for Nonconvex Tensor Completion: Confidence Intervals, Heteroscedasticity and Optimality

Abstract

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 twostage estimation algorithm proposed by [2], 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 2 accuracy-including both the rates and the pre-constants-when estimating both the unknown tensor and the underlying tensor factors.

Causal Inference for Social and Engineering Systems

“…In this example in Figure 4.6a, all users can be used as anchor rows, and the set of items that are commonly rated across all users can be used as anchor columns. Further, we remark that in this example P is not low-rank, thus violating the key assumption required to learn p ij in Ma and Chen (2019); Cai et al (2020); Bhattacharya and Chatterjee (2021).…”