# Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality

arXiv: https://arxiv.org/abs/2006.08580 · HTML: https://mlsanity.com/paper/2006.08580/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jun 2020.

## Built on by (most cited first, 13 of 13)

- [Causal Matrix Completion](https://mlsanity.com/paper/2109.15154/) (2021)
- [Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements](https://mlsanity.com/paper/2104.14526/) (2021)
- [Inference for Heteroskedastic PCA with Missing Data](https://mlsanity.com/paper/2107.12365/) (2021)
- [Provable Tensor-Train Format Tensor Completion by Riemannian Optimization](https://mlsanity.com/paper/2108.12163/) (2021)
- [Doubly robust nearest neighbors in factor models](https://mlsanity.com/paper/2211.14297/) (2022)
- [Deflated HeteroPCA: Overcoming the curse of ill-conditioning in heteroskedastic PCA](https://mlsanity.com/paper/2303.06198/) (2023)
- [Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descent](https://mlsanity.com/paper/2401.11940/) (2024)
- [Covariate-assisted Sparse Tensor Completion](https://mlsanity.com/paper/2103.06428/) (2021)
- [A non-asymptotic distributional theory of approximate message passing for sparse and robust regression](https://mlsanity.com/paper/2401.03923/) (2024)
- [Minimax Estimation of Linear Functions of Eigenvectors in the Face of Small Eigen-Gaps](https://mlsanity.com/paper/2104.03298/) (2021)
- [Jointly Modeling and Clustering Tensors in High Dimensions](https://mlsanity.com/paper/2104.07773/) (2021)
- [Tensor Methods in High Dimensional Data Analysis: Opportunities and Challenges](https://mlsanity.com/paper/2405.18412/) (2024)
- [Guaranteed Noisy CP Tensor Recovery via Riemannian Optimization on the Segre Manifold](https://mlsanity.com/paper/2510.00569/) (2025)

## Abstract (arXiv)

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 ℓ2 accuracy -- including both the rates and the pre-constants -- when estimating both the unknown tensor and the underlying tensor factors.