Efficient Sparse PCA via Block-Diagonalization

• URL: http://arxiv.org/abs/2410.14092v1
• Date: Fri, 18 Oct 2024 00:16:10 GMT
• Title: Efficient Sparse PCA via Block-Diagonalization
• Authors: Alberto Del Pia, Dekun Zhou, Yinglun Zhu,
• Abstract summary: We propose a novel framework to efficiently approximate Sparse PCA. It can leverage any off-the-shelf Sparse PCA algorithm and achieve significant computational speedups. Our framework, when integrated with this algorithm, reduces the runtime to $mathcalOleft(fracddstar cdot g(k, dstar) + d2right)$.
• Score: 13.38174941551702
• License:
• Abstract: Sparse Principal Component Analysis (Sparse PCA) is a pivotal tool in data analysis and dimensionality reduction. However, Sparse PCA is a challenging problem in both theory and practice: it is known to be NP-hard and current exact methods generally require exponential runtime. In this paper, we propose a novel framework to efficiently approximate Sparse PCA by (i) approximating the general input covariance matrix with a re-sorted block-diagonal matrix, (ii) solving the Sparse PCA sub-problem in each block, and (iii) reconstructing the solution to the original problem. Our framework is simple and powerful: it can leverage any off-the-shelf Sparse PCA algorithm and achieve significant computational speedups, with a minor additive error that is linear in the approximation error of the block-diagonal matrix. Suppose $g(k, d)$ is the runtime of an algorithm (approximately) solving Sparse PCA in dimension $d$ and with sparsity value $k$. Our framework, when integrated with this algorithm, reduces the runtime to $\mathcal{O}\left(\frac{d}{d^\star} \cdot g(k, d^\star) + d^2\right)$, where $d^\star \leq d$ is the largest block size of the block-diagonal matrix. For instance, integrating our framework with the Branch-and-Bound algorithm reduces the complexity from $g(k, d) = \mathcal{O}(k^3\cdot d^k)$ to $\mathcal{O}(k^3\cdot d \cdot (d^\star)^{k-1})$, demonstrating exponential speedups if $d^\star$ is small. We perform large-scale evaluations on many real-world datasets: for exact Sparse PCA algorithm, our method achieves an average speedup factor of 93.77, while maintaining an average approximation error of 2.15%; for approximate Sparse PCA algorithm, our method achieves an average speedup factor of 6.77 and an average approximation error of merely 0.37%.

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