# A Decomposition Algorithm for the Sparse Generalized Eigenvalue Problem

## TL;DR

This paper introduces a novel decomposition algorithm for the NP-hard sparse generalized eigenvalue problem, combining random, swapping, and combinatorial strategies to improve solution accuracy in statistical learning models.

## Contribution

It proposes a new decomposition method with theoretical analysis and demonstrates superior accuracy over existing solutions for the sparse generalized eigenvalue problem.

## Key findings

- Significantly outperforms existing solutions in accuracy
- Uses a combination of random, swapping, and combinatorial strategies
- Provides theoretical analysis of the proposed method

## Abstract

The sparse generalized eigenvalue problem arises in a number of standard and modern statistical learning models, including sparse principal component analysis, sparse Fisher discriminant analysis, and sparse canonical correlation analysis. However, this problem is difficult to solve since it is NP-hard. In this paper, we consider a new decomposition method to tackle this problem. Specifically, we use random or/and swapping strategies to find a working set and perform global combinatorial search over the small subset of variables. We consider a bisection search method and a coordinate descent method for solving the quadratic fractional programming subproblem. In addition, we provide some theoretical analysis for the proposed method. Our experiments have shown that the proposed method significantly and consistently outperforms existing solutions in term of accuracy.