  Regularized M-estimators are used in diverse areas of science and engineering
to fit high-dimensional models with some low-dimensional structure. Usually the
low-dimensional structure is encoded by the presence of the (unknown)
parameters in some low-dimensional model subspace. In such settings, it is
desirable for estimates of the model parameters to be \emph{model selection
consistent}: the estimates also fall in the model subspace. We develop a
general framework for establishing consistency and model selection consistency
of regularized M-estimators and show how it applies to some special cases of
interest in statistical learning. Our analysis identifies two key properties of
regularized M-estimators, referred to as geometric decomposability and
irrepresentability, that ensure the estimators are consistent and model
selection consistent.
