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 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.

Accepted in part to ICML 2020

Cited by in corpus (7)

• A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges
• Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements
• Provable Tensor-Train Format Tensor Completion by Riemannian Optimization
• Inference for Heteroskedastic PCA with Missing Data
• Subspace Estimation from Unbalanced and Incomplete Data Matrices: ${\ell }_{2,\infty }$ Statistical Guarantees
• Convex and Nonconvex Optimization Are Both Minimax-Optimal for Noisy Blind Deconvolution under Random Designs
• Implicit Regularization and Entrywise Convergence of Riemannian Optimization for Low Tucker-Rank Tensor Completion