Statistical Learning with Sparsity

226 SPARSE MULTIVARIATE METHODS

to the elastic net penalty in the special case Ω = γI. The resulting dis-

criminant vectors will be sparse if the regularization weight λ on the `

1

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penalty is sufficiently large. At the other extreme, if λ = 0, then minimizing

the criterion (8.42) is equivalent to the penalized discriminant analysis pro-

posal of Hastie et al. (1995). Although the criterion is nonconvex (due to the

quadratic constraints), a local optimum can be obtained via alternating min-

imization, using the elastic net to solve for β. In fact, if any convex penalties

are applied to the discriminant vectors in the optimal scoring criterion (8.41),

then it is easy to apply alternating minimization to solve the resulting prob-

lem. Moreover, there is ...