  Understanding how features interact with each other is of paramount
importance in many scientific discoveries and contemporary applications. Yet
interaction identification becomes challenging even for a moderate number of
covariates. In this paper, we suggest an efficient and flexible procedure,
called the interaction pursuit (IP), for interaction identification in
ultra-high dimensions. The suggested method first reduces the number of
interactions and main effects to a moderate scale by a new feature screening
approach, and then selects important interactions and main effects in the
reduced feature space using regularization methods. Compared to existing
approaches, our method screens interactions separately from main effects and
thus can be more effective in interaction screening. Under a fairly general
framework, we establish that for both interactions and main effects, the method
enjoys the sure screening property in screening and oracle inequalities in
selection. Our method and theoretical results are supported by several
simulation and real data examples.
