Predictive State Representations (PSRs), Observable Operator Models (OOMs) and Stochastic Multiplicity Automata (SMA) are basic types of models that have a common underlying algebraic structure. We briefly review this general structure and derive a generic learning framework into which the spectral learning algorithms fall. We introduce weights into the learning algorithm that reflect the accuracy of the estimates in the Hankel matrix, and show that this indeed improves the quality of the model estimation. Finally, we show that there is a close relationship between these weighted spectral learning algorithms and the efficiency sharpening algorithm for learning OOMs. The latter is an iterative algorithm that estimated the underlying subspace and weights from a previous model estimate, which has some computational advantages.