This work presents a comprehensive understanding of the estimation of a planted low-rank signal from a general spiked tensor model near the computational threshold. Relying on standard tools from the theory of large random matrices, we characterize the large-dimensional spectral behavior of the unfoldings of the data tensor and exhibit relevant signal-to-noise ratios governing the detectability of the principal directions of the signal. These results allow to accurately predict the reconstruction performance of truncated multilinear SVD (MLSVD) in the non-trivial regime. This is particularly important since it serves as an initialization of the higher-order orthogonal iteration (HOOI) scheme, whose convergence to the best low-multilinear-rank approximation depends entirely on its initialization. We give a sufficient condition for the convergence of HOOI and show that the number of iterations before convergence tends to $1$ in the large-dimensional limit.

翻译：本文系统性地研究了在计算阈值附近从一般尖峰张量模型中估计植入低秩信号的问题。借助大随机矩阵理论的标准工具，我们刻画了数据张量展开式的大维谱行为，并揭示了控制信号主方向可检测性的相关信噪比。这些结果能够精确预测在非平凡区域中截断多线性奇异值分解的重构性能。这一结论尤为重要，因为截断多线性奇异值分解可作为高阶正交迭代方案的初始化步骤，而高阶正交迭代收敛至最优低多线性秩近似完全取决于其初始化。我们给出了高阶正交迭代收敛的充分条件，并证明在大维极限下，收敛前的迭代次数趋近于1。