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 diverging 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. In theory 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. ROAD
selects an increasing number of features as the regularization 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.
Paper available at http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9868.2012.01029.x/abstract