On model selection consistency of regularized M-estimators

Penalized M-estimators are used in many areas of science and engineering to fit models with some low-dimensional structure in high-dimensional settings. In many problems arising in machine learning, signal processing, and high-dimensional statistics, the penalties are geometrically decomposable, i.e. can be expressed as a sum of support functions. We generalize the notion of irrepresentability and develop a general framework for establishing the model selection consistency of M-estimators with these penalties. We then use this framework to derive results for some special cases of interest in machine learning and high-dimensional statistics.