  In this paper, we study a new type of spatial sparse recovery problem, that
is to infer the fine-grained spatial distribution of certain density data in a
region only based on the aggregate observations recorded for each of its
subregions. One typical example of this spatial sparse recovery problem is to
infer spatial distribution of cellphone activities based on aggregate mobile
traffic volumes observed at sparsely scattered base stations. We propose a
novel Constrained Spatial Smoothing (CSS) approach, which exploits the local
continuity that exists in many types of spatial data to perform sparse recovery
via finite-element methods, while enforcing the aggregated observation
constraints through an innovative use of the ADMM algorithm. We also improve
the approach to further utilize additional geographical attributes. Extensive
evaluations based on a large dataset of phone call records and a demographical
dataset from the city of Milan show that our approach significantly outperforms
various state-of-the-art approaches, including Spatial Spline Regression (SSR).
