Solve sparse PCA problem by employing Hamiltonian system and leapfrog method

Principal Component Analysis (PCA) is a widely utilized technique for dimensionality reduction; however, its inherent lack of interpretability-stemming from dense linear combinations of all feature-limits its applicability in many domains. In this paper, we propose a novel sparse PCA algorithm that imposes sparsity through a smooth L1 penalty and leverages a Hamiltonian formulation solved via geometric integration techniques. Specifically, we implement two distinct numerical methods-one based on the Proximal Gradient (ISTA) approach and another employing a leapfrog (fourth-order Runge-Kutta) scheme-to minimize the energy function that balances variance maximization with sparsity enforcement. To extract a subset of sparse principal components, we further incorporate a deflation technique and subsequently transform the original high-dimensional face data into a lower-dimensional feature space. Experimental evaluations on a face recognition dataset-using both k-nearest neighbor and kernel ridge regression classifiers-demonstrate that the proposed sparse PCA methods consistently achieve higher classification accuracy than conventional PCA. Future research will extend this framework to integrate sparse PCA with modern deep learning architectures for multimodal recognition tasks.