On constrained and regularized high-dimensional regression

High-dimensional feature selection has become increasingly crucial for seeking parsimonious models in estimation. For selection consistency, we derive one necessary and sufficient condition formulated on the notion of degree of separation. The minimal degree of separation is necessary for any method to be selection consistent. At a level slightly higher than the minimal degree of separation, selection consistency is achieved by a constrained <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$L_0$$</EquationSource> </InlineEquation>-method and its computational surrogate—the constrained truncated <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$L_1$$</EquationSource> </InlineEquation>-method. This permits up to exponentially many features in the sample size. In other words, these methods are optimal in feature selection against any selection method. In contrast, their regularization counterparts—the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$L_0$$</EquationSource> </InlineEquation>-regularization and truncated <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$L_1$$</EquationSource> </InlineEquation>-regularization methods enable so under slightly stronger assumptions. More importantly, sharper parameter estimation/prediction is realized through such selection, leading to minimax parameter estimation. This, otherwise, is impossible in the absence of a good selection method for high-dimensional analysis. Copyright The Institute of Statistical Mathematics, Tokyo 2013