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An Augmented Lagrangian Approach for Sparse Principal Component Analysis
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.Comment: 42 page
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