Projection Algorithms for Non-Convex Minimization with Application to Sparse Principal Component Analysis

TL;DR

This paper introduces and analyzes projection algorithms for non-convex concave minimization problems, specifically applied to sparse principal component analysis, demonstrating competitive and sometimes superior performance over existing methods.

Contribution

It proposes gradient projection and approximate Newton algorithms with convergence proofs for non-convex sparse PCA problems, and compares their effectiveness with existing methods.

Findings

Gradient projection performs similarly to power methods.

Approximate Newton with BB Hessian can be faster and find better solutions.

Algorithms show promising results in numerical experiments.

Abstract

We consider concave minimization problems over non-convex sets.Optimization problems with this structure arise in sparse principal component analysis. We analyze both a gradient projection algorithm and an approximate Newton algorithm where the Hessian approximation is a multiple of the identity. Convergence results are established. In numerical experiments arising in sparse principal component analysis, it is seen that the performance of the gradient projection algorithm is very similar to that of the truncated power method and the generalized power method. In some cases, the approximate Newton algorithm with a Barzilai-Borwein (BB) Hessian approximation can be substantially faster than the other algorithms, and can converge to a better solution.