This paper investigates the estimation of low-rank multiple-input multiple-output (MIMO) channels in millimeter wave (mmWave) communications. Hybrid MIMO transceivers equipped with uniform linear arrays (ULAs) and phase shifter networks are considered. We propose a novel three-stage channel estimator by exploiting the low-rankness of the channel matrix and knowledge of the array response: We first obtain a low-rank estimate of the channel matrix using inductive matrix completion (IMC); then estimate the angle of arrival (AoA) and angle of departure (AoD) for the propagation paths by solving two one-dimensional spectrum estimation problems using Toeplitz rectification and the root multiple signal classification (MUSIC) algorithm; and finally pair the AoAs and AoDs and estimate the channel gains by solving a sparse recovery problem. Each stage is implemented at a low complexity and thus the overall complexity is kept low. A priori knowledge about the channel is exploited progressively to enhance the performance. The simulation results suggest significant gains in the channel estimation performance along with sparse representations of the estimated channel.