  Modern data acquisition routinely produces massive amounts of network data.
Though many methods and models have been proposed to analyze such data, the
research of network data is largely disconnected with the classical theory of
statistical learning and signal processing. In this paper, we present a new
framework for modeling network data, which connects two seemingly different
areas: network data analysis and compressed sensing. From a nonparametric
perspective, we model an observed network using a large dictionary. In
particular, we consider the network clique detection problem and show
connections between our formulation with a new algebraic tool, namely Randon
basis pursuit in homogeneous spaces. Such a connection allows us to identify
rigorous recovery conditions for clique detection problems. Though this paper
is mainly conceptual, we also develop practical approximation algorithms for
solving empirical problems and demonstrate their usefulness on real-world
datasets.
