Multi-Rank Sparse and Functional PCA: Manifold Optimization and Iterative Deflation Techniques
This work addresses computational and estimation challenges in high-dimensional data analysis for researchers and practitioners, though it is incremental as it builds on existing SFPCA methods.
The authors tackled the problem of estimating multiple principal components using the Sparse and Functional PCA (SFPCA) estimator by proposing a manifold optimization extension and iterative deflation techniques, resulting in significantly improved performance even when orthogonality constraints do not hold.
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 existing fast 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.