We investigate the power iteration algorithm for the tensor PCA model introduced in Richard and Montanari (2014). Previous work studying the properties of tensor power iteration is either limited to a constant number of iterations, or requires a non-trivial data-independent initialization. In this paper, we move beyond these limitations and analyze the dynamics of randomly initialized tensor power iteration up to polynomially many steps. Our contributions are threefold: First, we establish sharp bounds on the number of iterations required for power method to converge to the planted signal, for a broad range of the signal-to-noise ratios. Second, our analysis reveals that the actual algorithmic threshold for power iteration is smaller than the one conjectured in literature by a polylog(n) factor, where n is the ambient dimension. Finally, we propose a simple and effective stopping criterion for power iteration, which provably outputs a solution that is highly correlated with the true signal. Extensive numerical experiments verify our theoretical results.

翻译：我们研究了Richard和Montanari（2014）提出的张量PCA模型中的幂迭代算法。关于张量幂迭代性质的现有工作要么局限于常数次迭代，要么需要非平凡的数据无关初始化。本文突破了这些局限，分析了随机初始化的张量幂迭代在多项式步数内的动力学特性。我们的贡献有三重：首先，对于广泛的信噪比范围，我们建立了幂方法收敛至植入信号所需迭代次数的锐利界。其次，我们的分析揭示幂迭代的实际算法阈值比文献中推测的要小一个polylog(n)因子（其中n为环境维度）。最后，我们提出了一种简单有效的幂迭代停止准则，该准则可证明输出与真实信号高度相关的解。大量数值实验验证了我们的理论结果。