Alternating direction method of multipliers for penalized zero-variance discriminant analysis

TL;DR

This paper introduces SZVD, a new method combining zero-variance discriminant analysis with sparsity-inducing penalties, optimized via ADMM, for high-dimensional classification with theoretical convergence guarantees.

Contribution

It proposes a novel heuristic for high-dimensional classification that integrates feature selection with discriminant analysis using a nonconvex optimization approach and ADMM.

Findings

Effective in classifying simulated high-dimensional data

Demonstrates practical utility on time-series datasets

Provides convergence guarantees for the proposed algorithm

Abstract

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.