Title: Statistical Limits in Random Tensors with Multiple Correlated Spikes Show Chinese title

Title: 随机张量中多个相关尖峰的统计极限

Abstract: We use tools from random matrix theory to study the multi-spiked tensor model, i.e., a rank-$r$ deformation of a symmetric random Gaussian tensor. In particular, thanks to the nature of local optimization methods used to find the maximum likelihood estimator of this model, we propose to study the phase transition phenomenon for finding critical points of the corresponding optimization problem, i.e., those points defined by the Karush-Kuhn-Tucker (KKT) conditions. Moreover, we characterize the limiting alignments between the estimated signals corresponding to a critical point of the likelihood and the ground truth signals. With the help of these results, we propose a new estimator of the rank-$r$ tensor weights by solving a system of polynomial equations, which is asymptotically unbiased contrary the maximum likelihood estimator. Show Chinese abstract

Abstract: 我们使用随机矩阵理论的工具来研究多尖峰张量模型，即对称随机高斯张量的秩-$r$变形。 特别是，由于用于寻找该模型最大似然估计量的局部优化方法的性质，我们提出研究对应优化问题的临界点的相变现象，即由Karush-Kuhn-Tucker (KKT)条件定义的那些点。 此外，我们描述了对应于似然函数临界点的估计信号与真实信号之间的极限对齐情况。 借助这些结果，我们通过求解多项式方程组提出了一种新的秩-$r$张量权重估计量，该估计量在渐近意义上是无偏的，而最大似然估计量则不是。