Uncertainty Quantification for Noisy Low-tubal-rank Tensor Completion

TL;DR

This paper develops a statistical inference framework for low-tubal-rank tensor completion, enabling uncertainty quantification and hypothesis testing for high-dimensional tensor data.

Contribution

It introduces a novel inference method using double-sample debiasing and low-rank projection to produce asymptotically Gaussian estimators for tensors.

Findings

The proposed method provides valid confidence intervals and hypothesis tests.

Simulations confirm the theoretical guarantees and robustness.

Application to geophysical data demonstrates practical effectiveness.

Abstract

High-dimensional tensor data often exhibit strong temporal correlations that appear as low-dimensional structures in the frequency domain. While the low-tubal-rank tensor model effectively captures these spectral features, making it potentially suitable for geophysical data, existing methods primarily focus on point estimation. Uncertainty quantification (UQ) of imputed values and rigorous statistical inference for these models remain largely unexplored. In this work, we propose a flexible inference framework for linear forms of high-dimensional tensors. Employing a double-sample debiasing technique followed by a low-rank projection, we construct asymptotically Gaussian estimators that yield valid statistical inference under mild assumptions. More precisely, we can perform hypothesis testing and construct confidence intervals with this result. We validate the theoretical results through extensive simulations and demonstrate the practical effectiveness of our method in completing the global total electron content data. We demonstrate, using those numerical results, that our entrywise confidence intervals are robust and reliable, yielding informative uncertainty quantification that captures underlying variability.