Statistical Inference in Tensor Completion: Optimal Uncertainty Quantification and Statistical-Computational Gaps

This paper presents a simple yet efficient method for statistical inference of tensor linear forms with incomplete and noisy observations. Under the Tucker low-rank tensor model, we utilize an appropriate initial estimate, along with a debiasing technique followed by a one-step power iteration, to construct an asymptotic normal test statistic. This method is suitable for various statistical inference tasks, including confidence interval prediction, inference under heteroskedastic and sub-exponential noises, and simultaneous testing. Furthermore, the approach reaches the Cram\'er-Rao lower bound for statistical estimation on Riemannian manifolds, indicating its optimality for uncertainty quantification. We comprehensively discusses the statistical-computational gaps and investigates the relationship between initialization and bias-correlation approaches. The findings demonstrate that with independent initialization, statistically optimal sample sizes and signal-to-noise ratios are sufficient for accurate inferences. Conversely, when initialization depends on the observations, computationally optimal sample sizes and signal-to-noise ratios also guarantee asymptotic normality without the need for data-splitting. The phase transition of computational and statistical limits is presented. Numerical simulations results conform to the theoretical discoveries.