  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.
