Alternating direction method of multipliers for penalized zero-variance discriminant analysis
This is an incremental method for classification in high-dimensional settings, addressing feature selection and sparsity for researchers in machine learning and statistics.
The paper tackles classification in high-dimensional data by proposing sparse zero-variance discriminant analysis (SZVD), a heuristic that combines linear discriminant analysis with feature selection, and develops an algorithm based on the alternating direction method of multipliers to solve it, showing empirical effectiveness on simulated and time-series data.
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.