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.

翻译：本文针对含噪声的不完整观测数据，提出了一种简洁高效的张量线性形式统计推断方法。在Tucker低秩张量模型框架下，我们采用适当的初始估计，结合去偏技术与一步幂迭代，构建了渐近正态检验统计量。该方法适用于多种统计推断任务，包括置信区间预测、异方差与次指数噪声下的推断以及同步检验。此外，该方法达到了黎曼流形上统计估计的克拉默-拉奥下界，表明其在不确定性量化方面具有最优性。我们系统探讨了统计-计算间隙，并深入研究了初始化方法与偏差校正策略之间的关联。研究结果表明：采用独立初始化时，达到统计最优的样本量与信噪比即可保证推断准确性；而当初始化依赖于观测数据时，计算最优的样本量与信噪比同样能确保渐近正态性，且无需数据分割操作。本文揭示了计算极限与统计极限的相变现象，数值模拟结果与理论发现高度吻合。