We study the dynamics of gradient flow in high dimensions for the multi-spiked tensor problem, where the goal is to estimate $r$ unknown signal vectors (spikes) from noisy Gaussian tensor observations. Specifically, we analyze the maximum likelihood estimation procedure, which involves optimizing a highly nonconvex random function. We determine the sample complexity required for gradient flow to efficiently recover all spikes, without imposing any assumptions on the separation of the signal-to-noise ratios (SNRs). More precisely, our results provide the sample complexity required to guarantee recovery of the spikes up to a permutation. Our work builds on our companion paper [Ben Arous, Gerbelot, Piccolo 2024], which studies Langevin dynamics and determines the sample complexity and separation conditions for the SNRs necessary for ensuring exact recovery of the spikes (where the recovered permutation matches the identity). During the recovery process, the correlations between the estimators and the hidden vectors increase in a sequential manner. The order in which these correlations become significant depends on their initial values and the corresponding SNRs, which ultimately determines the permutation of the recovered spikes.

翻译：我们研究了多尖峰张量问题中梯度流在高维空间中的动力学行为，该问题的目标是从含噪高斯张量观测中估计$r$个未知信号向量（尖峰）。具体而言，我们分析了最大似然估计过程，该过程涉及对高度非凸随机函数的优化。我们确定了梯度流有效恢复所有尖峰所需的样本复杂度，且未对信噪比（SNRs）的分离程度施加任何假设。更精确地说，我们的研究结果提供了保证尖峰恢复（至置换等价）所需的样本复杂度。本工作基于我们的姊妹论文[Ben Arous, Gerbelot, Piccolo 2024]，该论文研究了朗之万动力学，并确定了确保尖峰精确恢复（恢复置换与恒等置换匹配）所需的样本复杂度及SNRs分离条件。在恢复过程中，估计量与隐藏向量之间的相关性以顺序方式递增。这些相关性变得显著的时间顺序取决于其初始值及相应的SNRs，这最终决定了恢复尖峰的置换顺序。