Penalized Orthogonal Iteration for Sparse Estimation of Generalized Eigenvalue Problem

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.

References in corpus (6)

• Sparse principal component analysis and iterative thresholding
• PCA consistency in high dimension, low sample size context
• Coordinate-independent sparse sufficient dimension reduction and variable selection
• Sparse Generalized Eigenvalue Problem via Smooth Optimization
• Simultaneous sparse estimation of canonical vectors in the p>>N setting
• Penalized versus constrained generalized eigenvalue problems

Cited by in corpus (2)

• Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA)
• Variable selection and basis learning for ordinal classification