A Decomposition Framework for Certifiably Optimal Orthogonal Sparse PCA

• URL: http://arxiv.org/abs/2603.01144v1
• Date: Sun, 01 Mar 2026 15:09:37 GMT
• Title: A Decomposition Framework for Certifiably Optimal Orthogonal Sparse PCA
• Authors: Difei Cheng, Qiao Hu,
• Abstract summary: We introduce a novel Sparse Principal Component Analysis (SPCA) algorithm called textscGS-SPCA (SPCA with Gram-Schmidt Orthogonalization)<n>By incorporating this strategy, we are able to obtain $varepsilon$-optimal solutions with a trade-off between precision and efficiency.
• Score: 1.8252316736244685
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Sparse Principal Component Analysis (SPCA) is an important technique for high-dimensional data analysis, improving interpretability by imposing sparsity on principal components. However, existing methods often fail to simultaneously guarantee sparsity, orthogonality, and optimality of the principal components. To address this challenge, this work introduces a novel Sparse Principal Component Analysis (SPCA) algorithm called \textsc{GS-SPCA} (SPCA with Gram-Schmidt Orthogonalization), which simultaneously enforces sparsity, orthogonality, and optimality. However, the original GS-SPCA algorithm is computationally expensive due to the inherent $\ell_0$-norm constraint. To address this issue, we propose two acceleration strategies: First, we combine \textbf{Branch-and-Bound} with the GS-SPCA algorithm. By incorporating this strategy, we are able to obtain $\varepsilon$-optimal solutions with a trade-off between precision and efficiency, significantly improving computational speed. Second, we propose a \textbf{decomposition framework} for efficiently solving \textbf{multiple} principal components. This framework approximates the covariance matrix using a block-diagonal matrix through a thresholding method, reducing the original SPCA problem to a set of block-wise subproblems on approximately block-diagonal matrices.

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