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

TL;DR

This paper develops an efficient method for statistical inference in tensor completion under low-rank models, achieving optimal uncertainty quantification and exploring the gap between statistical and computational limits.

Contribution

It introduces a debiasing and one-step power iteration approach for tensor inference that attains the Cramér-Rao bound and analyzes the impact of initialization on sample complexity.

Findings

Estimator achieves the Cramér-Rao lower bound.

Statistically optimal sample sizes suffice with independent initialization.

Computationally optimal conditions are sufficient with dependent initialization.

Abstract

This paper presents a simple yet efficient method for statistical inference of tensor linear forms using incomplete and noisy observations. Under the Tucker low-rank tensor model and the missing-at-random assumption, we utilize an appropriate initial estimate along with a debiasing technique followed by a one-step power iteration to construct an asymptotically normal test statistic. This method is suitable for various statistical inference tasks, including constructing confidence intervals, inference under heteroskedastic and sub-exponential noise, and simultaneous testing. We demonstrate that the estimator achieves the Cram\'er-Rao lower bound on Riemannian manifolds, indicating its optimality in uncertainty quantification. We comprehensively examine the statistical-to-computational gaps and investigate the impact of initialization on the minimal conditions regarding sample size and signal-to-noise ratio required for accurate inference. Our findings show that with independent initialization, statistically optimal sample sizes and signal-to-noise ratios are sufficient for accurate inference. Conversely, if only dependent initialization is available, computationally optimal sample sizes and signal-to-noise ratio conditions still guarantee asymptotic normality without the need for data-splitting. We present the phase transition between computational and statistical limits. Numerical simulation results align with the theoretical findings.