  We study the problem of selecting a subset of k random variables from a large
set, in order to obtain the best linear prediction of another variable of
interest. This problem can be viewed in the context of both feature selection
and sparse approximation. We analyze the performance of widely used greedy
heuristics, using insights from the maximization of submodular functions and
spectral analysis. We introduce the submodularity ratio as a key quantity to
help understand why greedy algorithms perform well even when the variables are
highly correlated. Using our techniques, we obtain the strongest known
approximation guarantees for this problem, both in terms of the submodularity
ratio and the smallest k-sparse eigenvalue of the covariance matrix. We further
demonstrate the wide applicability of our techniques by analyzing greedy
algorithms for the dictionary selection problem, and significantly improve the
previously known guarantees. Our theoretical analysis is complemented by
experiments on real-world and synthetic data sets; the experiments show that
the submodularity ratio is a stronger predictor of the performance of greedy
algorithms than other spectral parameters.
