  For high-dimensional classification, it is well known that naively performing
the Fisher discriminant rule leads to poor results due to diverging spectra and
noise accumulation. Therefore, researchers proposed independence rules to
circumvent the diverse spectra, and sparse independence rules to mitigate the
issue of noise accumulation. However, in biological applications, there are
often a group of correlated genes responsible for clinical outcomes, and the
use of the covariance information can significantly reduce misclassification
rates. The extent of such error rate reductions is unveiled by comparing the
misclassification rates of the Fisher discriminant rule and the independence
rule. To materialize the gain based on finite samples, a Regularized Optimal
Affine Discriminant (ROAD) is proposed based on a covariance penalty. ROAD
selects an increasing number of features as the penalization relaxes. Further
benefits can be achieved when a screening method is employed to narrow the
feature pool before hitting the ROAD. An efficient Constrained Coordinate
Descent algorithm (CCD) is also developed to solve the associated optimization
problems. Sampling properties of oracle type are established. Simulation
studies and real data analysis support our theoretical results and demonstrate
the advantages of the new classification procedure under a variety of
correlation structures. A delicate result on continuous piecewise linear
solution path for the ROAD optimization problem at the population level
justifies the linear interpolation of the CCD algorithm.
