Projection Algorithms for Non-Convex Minimization with Application to Sparse Principal Component Analysis
For researchers in sparse PCA and non-convex optimization, this work provides a faster algorithm with improved convergence, though the improvements are incremental over existing methods.
The paper analyzes gradient projection and approximate Newton algorithms for concave minimization over non-convex sets, applied to sparse PCA. The approximate Newton method with Barzilai-Borwein steps is substantially faster and converges to better solutions than the truncated power method and generalized power method in some cases.
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.