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

## TL;DR

This paper introduces advanced methods for estimating multiple sparse and functional principal components using manifold optimization and iterative deflation, improving efficiency and performance over existing techniques.

## Contribution

It extends SFPCA to multiple components with manifold optimization and develops efficient deflation algorithms for better estimation.

## Key findings

- Manifold optimization effectively estimates multiple components.
- Iterative deflation improves computational efficiency.
- Both methods outperform traditional approaches in non-orthogonal scenarios.

## Abstract

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.