Truncated Power Method for Sparse Eigenvalue Problems

TL;DR

This paper introduces the truncated power method, an effective algorithm for solving sparse eigenvalue problems, with strong theoretical guarantees and competitive performance in applications like sparse PCA and subgraph detection.

Contribution

The paper presents a novel truncated power method with proven sparse recovery guarantees for nonconvex sparse eigenvalue problems.

Findings

Proven strong sparse recovery results for the method.

Competitive empirical performance on large-scale datasets.

Effective application to sparse PCA and densest subgraph problems.

Abstract

This paper considers the sparse eigenvalue problem, which is to extract dominant (largest) sparse eigenvectors with at most $k$ non-zero components. We propose a simple yet effective solution called truncated power method that can approximately solve the underlying nonconvex optimization problem. A strong sparse recovery result is proved for the truncated power method, and this theory is our key motivation for developing the new algorithm. The proposed method is tested on applications such as sparse principal component analysis and the densest $k$-subgraph problem. Extensive experiments on several synthetic and real-world large scale datasets demonstrate the competitive empirical performance of our method.