Multi-Rank Sparse and Functional PCA: Manifold Optimization and Iterative Deflation Techniques

We consider the problem of estimating multiple principal components using the recently-proposed Sparse and Functional Principal Components Analysis (SFPCA) estimator. We first propose an extension of SFPCA which estimates several principal components simultaneously using manifold optimization techniques to enforce orthogonality constraints. While effective, this approach is computationally burdensome so we also consider iterative deflation approaches which take advantage of efficient algorithms for rank-one SFPCA. We show that alternative deflation schemes can more efficiently extract signal from the data, in turn improving estimation of subsequent components. Finally, we compare the performance of our manifold optimization and deflation techniques in a scenario where orthogonality does not hold and find that they still lead to significantly improved performance.