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%.

翻译：稀疏主成分分析（Sparse PCA）是数据分析和降维中的关键工具。然而，稀疏主成分分析在理论和实践上都是一个具有挑战性的问题：已知它是NP难问题，且当前精确方法通常需要指数级运行时间。本文提出一种新颖框架，通过以下步骤高效逼近稀疏主成分分析：（i）用重排序的块对角矩阵逼近一般输入协方差矩阵，（ii）在每个块中求解稀疏主成分分析子问题，以及（iii）重构原始问题的解。我们的框架简单而强大：它可以利用任何现成的稀疏主成分分析算法，并实现显著的计算加速，其附加误差仅与块对角矩阵的逼近误差呈线性关系。假设 $g(k, d)$ 是（近似）求解维度 $d$、稀疏度 $k$ 的稀疏主成分分析算法的运行时间。我们的框架与该算法结合后，可将运行时间降低至 $\mathcal{O}\left(\frac{d}{d^\star} \cdot g(k, d^\star) + d^2\right)$，其中 $d^\star \leq d$ 是块对角矩阵的最大块尺寸。例如，将我们的框架与分支定界算法结合，可将复杂度从 $g(k, d) = \mathcal{O}(k^3\cdot d^k)$ 降低至 $\mathcal{O}(k^3\cdot d \cdot (d^\star)^{k-1})$，若 $d^\star$ 较小，则展现出指数级加速。我们在多个真实世界数据集上进行了大规模评估：对于精确稀疏主成分分析算法，我们的方法实现了平均93.77倍的加速，同时保持平均2.15%的逼近误差；对于近似稀疏主成分分析算法，我们的方法实现了平均6.77倍的加速，且平均逼近误差仅为0.37%。