  We consider the task of classification in the high dimensional setting where
the number of features of the given data is significantly greater than the
number of observations. To accomplish this task, we propose a heuristic, called
sparse zero-variance discriminant analysis (SZVD), for simultaneously
performing linear discriminant analysis and feature selection on high
dimensional data. This method combines classical zero-variance discriminant
analysis, where discriminant vectors are identified in the null space of the
sample within-class covariance matrix, with penalization applied to induce
sparse structures in the resulting vectors. To approximately solve the
resulting nonconvex problem, we develop a simple algorithm based on the
alternating direction method of multipliers. Further, we show that this
algorithm is applicable to a larger class of penalized generalized eigenvalue
problems, including a particular relaxation of the sparse principal component
analysis problem. Finally, we establish theoretical guarantees for convergence
of our algorithm to stationary points of the original nonconvex problem, and
empirically demonstrate the effectiveness of our heuristic for classifying
simulated data and data drawn from applications in time-series classification.

Machine Learning;Optimization and Control