Proximal Methods for Sparse Optimal Scoring and Discriminant Analysis

Abstract

Linear discriminant analysis (LDA) is a classical method for dimensionality reduction, where discriminant vectors are sought to project data to a lower dimensional space for optimal separability of classes. Several recent papers have outlined strategies for exploiting sparsity for using LDA with high-dimensional data. However, many lack scalable methods for solution of the underlying optimization problems. We propose three new numerical optimization schemes for solving the sparse optimal scoring formulation of LDA based on block coordinate descent, the proximal gradient method, and the alternating direction method of multipliers. We show that the per-iteration cost of these methods scales linearly in the dimension of the data provided restricted regularization terms are employed, and cubically in the dimension of the data in the worst case. Furthermore, we establish that if our block coordinate descent framework generates convergent subsequences of iterates, then these subsequences converge to the stationary points of the sparse optimal scoring problem. We demonstrate the effectiveness of our new methods with empirical results for classification of Gaussian data and data sets drawn from benchmarking repositories, including time-series and multispectral X-ray data, and provide Matlab and R implementations of our optimization schemes.