We study the dynamics of two local optimization algorithms, online stochastic gradient descent (SGD) and gradient flow, within the framework of the multi-spiked tensor model in the high-dimensional regime. This multi-index model arises from the tensor principal component analysis (PCA) problem, which aims to infer $r$ unknown, orthogonal signal vectors within the $N$-dimensional unit sphere through maximum likelihood estimation from noisy observations of an order-$p$ tensor. We determine the number of samples and the conditions on the signal-to-noise ratios (SNRs) required to efficiently recover the unknown spikes from natural initializations. Specifically, we distinguish between three types of recovery: exact recovery of each spike, recovery of a permutation of all spikes, and recovery of the correct subspace spanned by the signal vectors. We show that with online SGD, it is possible to recover all spikes provided a number of sample scaling as $N^{p-2}$, aligning with the computational threshold identified in the rank-one tensor PCA problem [Ben Arous, Gheissari, Jagannath 2020, 2021]. For gradient flow, we show that the algorithmic threshold to efficiently recover the first spike is also of order $N^{p-2}$. However, recovering the subsequent directions requires the number of samples to scale as $N^{p-1}$. Our results are obtained through a detailed analysis of a low-dimensional system that describes the evolution of the correlations between the estimators and the spikes. In particular, the hidden vectors are recovered one by one according to a sequential elimination phenomenon: as one correlation exceeds a critical threshold, all correlations sharing a row or column index decrease and become negligible, allowing the subsequent correlation to grow and become macroscopic. The sequence in which correlations become macroscopic depends on their initial values and on the associated SNRs.

本文研究了在高维模式下，两个局部优化算法（在线随机梯度下降（SGD）和梯度流）在多尖峰张量模型中的动态特性。研究指出，通过合适的样本数量和信噪比条件，这些算法能够有效恢复未知的信号向量，其中在线SGD实现所有尖峰的精确恢复，而对于梯度流，恢复后续方向所需的样本数量则更高。此研究为张量主成分分析领域提供了重要的理论基础。

高维优化的多尖峰张量主成分分析