Penalized Orthogonal Iteration for Sparse Estimation of Generalized Eigenvalue Problem

TL;DR

This paper introduces a penalized orthogonal iteration algorithm for sparse estimation of eigenvectors in generalized eigenvalue problems, applicable to PCA, LDA, CCA, and SDR, improving accuracy and efficiency.

Contribution

The paper develops a novel penalized orthogonal iteration method that generalizes eigenvector estimation with sparsity, offering a more accurate and computationally efficient approach.

Findings

The proposed algorithms outperform existing methods in accuracy.

The tuning procedure effectively selects sparsity levels.

Applications demonstrate successful sparse estimation in various statistical models.

Abstract

We propose a new algorithm for sparse estimation of eigenvectors in generalized eigenvalue problems (GEP). The GEP arises in a number of modern data-analytic situations and statistical methods, including principal component analysis (PCA), multiclass linear discriminant analysis (LDA), canonical correlation analysis (CCA), sufficient dimension reduction (SDR) and invariant co-ordinate selection. We propose to modify the standard generalized orthogonal iteration with a sparsity-inducing penalty for the eigenvectors. To achieve this goal, we generalize the equation-solving step of orthogonal iteration to a penalized convex optimization problem. The resulting algorithm, called penalized orthogonal iteration, provides accurate estimation of the true eigenspace, when it is sparse. Also proposed is a computationally more efficient alternative, which works well for PCA and LDA problems. Numerical studies reveal that the proposed algorithms are competitive, and that our tuning procedure works well. We demonstrate applications of the proposed algorithm to obtain sparse estimates for PCA, multiclass LDA, CCA and SDR. Supplementary materials are available online.