  We propose a method for variable selection and basis learning for
high-dimensional classification with ordinal responses. The proposed method
extends sparse multiclass linear discriminant analysis, with the aim of
identifying not only the variables relevant to discrimination but also the
variables that are order-concordant with the responses. For this purpose, we
compute for each variable an ordinal weight, where larger weights are given to
variables with ordered group-means, and penalize the variables with smaller
weights more severely in the proposed sparse basis learning. A two-step
construction for ordinal weights is developed, and we show that the ordinal
weights correctly separate ordinal variables from non-ordinal variables with
high probability. The resulting sparse ordinal basis learning method is shown
to consistently select either the discriminant variables or the ordinal and
discriminant variables, depending on the choice of a tunable parameter. Such
asymptotic guarantees are given under a high-dimensional asymptotic regime
where the dimension grows much faster than the sample size. Simulated and real
data analyses further confirm that the proposed basis learning provides more
sparse basis, mostly consisting of ordinal variables, than other basis learning
methods, including those developed for ordinal classification.
