We present two reduced-rank channel estimators for large-scale multiple-input, multiple-output (MIMO) systems and analyze their mean square error (MSE) performance. Taking advantage of the channel's transform domain sparseness, the estimators yield outstanding performance and may offer additional mean angle-of-arrival (AoA) information. It is shown that, for the estimators to be effective, one has to select a proper predetermined unitary basis (transform) and be able to determine the dominant channel rank and the associated subspace. Further MSE analysis reveals the relations among the array size, channel rank, signal-to-noise ratio (SNR), and the estimators' performance. It provides rationales for the proposed rank determination and mean AoA estimation algorithms as well. An angle alignment operation included in one of our channel models is proved to be effective in further reducing the required rank, shifting the dominant basis vectors' range (channel spectrum) and improving the estimator's performance when a suitable basis is used. We also draw insightful analogies among MIMO channel modeling, transform coding, parallel spatial search, and receive beamforming. Computer experiment results are provided to examine the numerical effects of various estimator parameters and the advantages of the proposed channel estimators and rank determination method.