  We propose a new penalized method for variable selection and estimation that
explicitly incorporates the correlation patterns among predictors. This method
is based on a combination of the minimax concave penalty and Laplacian
quadratic associated with a graph as the penalty function. We call it the
sparse Laplacian shrinkage (SLS) method. The SLS uses the minimax concave
penalty for encouraging sparsity and Laplacian quadratic penalty for promoting
smoothness among coefficients associated with the correlated predictors. The
SLS has a generalized grouping property with respect to the graph represented
by the Laplacian quadratic. We show that the SLS possesses an oracle property
in the sense that it is selection consistent and equal to the oracle Laplacian
shrinkage estimator with high probability. This result holds in sparse,
high-dimensional settings with p >> n under reasonable conditions. We derive a
coordinate descent algorithm for computing the SLS estimates. Simulation
studies are conducted to evaluate the performance of the SLS method and a real
data example is used to illustrate its application.
