In an age of exponentially increasing data generation, performing inference tasks by utilizing the available information in its entirety is not always an affordable option. The present paper puts forth approaches to render tracking of large-scale dynamic processes via a Kalman filter affordable, by processing a reduced number of data. Three distinct methods are introduced for reducing the number of data involved in the correction step of the filter. Towards this goal, the first two methods employ random projections and innovation-based censoring to effect dimensionality reduction and measurement selection respectively. The third method achieves reduced complexity by leveraging sequential processing of observations and selecting a few informative updates based on an information-theoretic metric. Simulations on synthetic data, compare the proposed methods with competing alternatives, and corroborate their efficacy in terms of estimation accuracy over complexity reduction. Finally, monitoring large networks is considered as an application domain, with the proposed methods tested on Kronecker graphs to evaluate their efficiency in tracking traffic matrices and time-varying link costs.