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：本文系统研究了在计算阈值附近，从一般尖峰张量模型中估计植入的低秩信号问题。依托大随机矩阵理论的标准工具，我们刻画了数据张量展开在大维尺度下的谱行为，并揭示了控制信号主方向可检测性的相关信噪比。这些结果能够准确预测非平凡区域中截断多线性奇异值分解的重构性能。由于该分解作为高阶正交迭代方案的初始化步骤，其收敛至最佳低多线性秩逼近完全取决于初始化的质量，因此这一发现尤为重要。我们给出了HOOI收敛的充分条件，并证明在大维极限下，该算法收敛前的迭代次数趋于1。