Statistical inference in tensor completion: Optimal uncertainty quantification and statistical-to-computational gaps
研究利用不完整且有噪声的观测数据对张量线性形式做统计推断的方法，通过去偏和一步幂迭代得到渐近正态的检验统计量，达到克拉美罗下界，适用于置信区间和多重检验。
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 Cramér–Rao lower bound on Riemannian manifolds, indicating its optimality in uncertainty quantification. We comprehensively examine statistical-to-computational gaps and investigate the impact of initialization on the minimal conditions regarding sample sizes and signal-to-noise ratios 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 ratios 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.