Principal component analysis (PCA) is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering. However, the standard PCA suffers from the fact that the principal components (PCs) are usually linear combinations of all the original variables, and it is thus often difficult to interpret the PCs. To alleviate this drawback, various sparse PCA approaches were proposed in literature [15, 6, 17, 28, 8, 25, 18, 7, 16]. Despite success in achieving sparsity, some important properties enjoyed by the standard PCA are lost in these methods such as uncorrelation of PCs and orthogonality of loading vectors. Also, the total explained variance that they attempt to maximize can be too optimistic. In this paper we propose a new formulation for sparse PCA, aiming at finding sparse and nearly uncorrelated PCs with orthogonal loading vectors while explaining as much of the total variance as possible. We also develop a novel augmented Lagrangian method for solving a class of nonsmooth constrained optimization problems, which is well suited for our formulation of sparse PCA. We show that it converges to a feasible point, and moreover under some regularity assumptions, it converges to a stationary point. Additionally, we propose two nonmonotone gradient methods for solving the augmented Lagrangian subproblems, and establish their global and local convergence. Finally, we compare our sparse PCA approach with several existing methods on synthetic, random, and real data, respectively. The computational results demonstrate that the sparse PCs produced by our approach substantially outperform those by other methods in terms of total explained variance, correlation of PCs, and orthogonality of loading vectors.

本研究提出了一种新的稀疏PCA方法，旨在找到稀疏和几乎不相关的主成分，并具有正交的载荷向量，同时尽可能多地解释总方差。我们还开发了一种新的增广Lagrangian方法来解决一类非光滑约束优化问题，该方法非常适合我们的稀疏PCA公式。最后，我们将我们的稀疏PCA方法与其他方法在合成数据，随机数据和真实数据上进行比较。计算结果表明，我们的方法产生的稀疏主成分在总方差，主成分相关性和载荷向量的正交性等方面显着优于其他方法。

稀疏主成分分析的增广拉格朗日方法