Penalized Orthogonal Iteration for Sparse Estimation of Generalized Eigenvalue Problem

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

Spectral Generalized Multi-Dimensional Scaling

Multidimensional scaling (MDS) is a family of methods that embed a given set of points into a simple, usually flat, domain. The points are assumed to be sampled from some metric space, and the mapping attempts to preserve the distances between each pair of points in the set. Distances in the target space can be computed analytically in this setting. Generalized MDS is an extension that allows mapping one metric space into another, that is, multidimensional scaling into target spaces in which distances are evaluated numerically rather than analytically. Here, we propose an efficient approach for computing such mappings between surfaces based on their natural spectral decomposition, where the surfaces are treated as sampled metric-spaces. The resulting spectral-GMDS procedure enables efficient embedding by implicitly incorporating smoothness of the mapping into the problem, thereby substantially reducing the complexity involved in its solution while practically overcoming its non-convex nature. The method is compared to existing techniques that compute dense correspondence between shapes. Numerical experiments of the proposed method demonstrate its efficiency and accuracy compared to state-of-the-art approaches