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.

翻译：本文提出了一种简单而高效的方法，用于利用不完整且有噪声的观测数据对张量线性形式进行统计推断。在Tucker低秩张量模型和随机缺失假设下，我们采用适当的初始估计，结合去偏技术和一步幂迭代，构建出渐近正态的检验统计量。该方法适用于多种统计推断任务，包括构建置信区间、异方差与次指数噪声下的推断以及同时检验。我们证明了该估计量在黎曼流形上达到了Cramér-Rao下界，表明其在不确定性量化方面具有最优性。我们系统考察了统计-计算间隙，并研究了初始化对精确推断所需样本量与信噪比最低条件的影响。研究结果表明：采用独立初始化时，统计最优的样本量与信噪比即可实现精确推断；反之，若仅能获得依赖初始化，计算最优的样本量与信噪比条件仍能保证渐近正态性，且无需数据分割。我们揭示了计算极限与统计极限之间的相变现象，数值模拟结果与理论发现一致。