  We consider the problem of model selection and estimation in sparse high
dimensional linear regression models with strongly correlated variables. First,
we study the theoretical properties of the dual Lasso solution, and we show
that joint consideration of the Lasso primal and its dual solutions are useful
for selecting correlated active variables. Second, we argue that correlations
among active predictors are not problematic, and we derive a new weaker
condition on the design matrix, called Pseudo Irrepresentable Condition (PIC).
Third, we present a new variable selection procedure, Dual Lasso Selector, and
we prove that the PIC is a necessary and sufficient condition for consistent
variable selection for the proposed method. Finally, by combining the dual
Lasso selector further with the Ridge estimation even better prediction
performance is achieved. We call the combination (DLSelect+Ridge), it can be
viewed as a new combined approach for inference in high-dimensional regression
models with correlated variables. We illustrate DLSelect+Ridge method and
compare it with popular existing methods in terms of variable selection,
prediction accuracy, estimation accuracy and computation speed by considering
various simulated and real data examples.
