Penalized Euclidean Distance Regression

A new method is proposed for variable screening, variable selection and prediction in linear regression problems where the number of predictors can be much larger than the number of observations. The method involves minimizing a penalized Euclidean distance, where the penalty is the geometric mean of the $\ell_1$ and $\ell_2$ norms of the regression coefficients. This particular formulation exhibits a grouping effect, which is useful for screening out predictors in higher or ultra-high dimensional problems. Also, an important result is a signal recovery theorem, which does not require an estimate of the noise standard deviation. Practical performances of variable selection and prediction are evaluated through simulation studies and the analysis of a dataset of mass spectrometry scans from melanoma patients, where excellent predictive performance is obtained.