PCA Sparsified
提出一种反向解决稀疏主成分分析问题的方法，通过最小化载荷非零元素个数并满足方差约束，使用半定规划和迭代加权ℓ1范数最小化算法，数值实验显示其有效性。
.We propose an inverted approach to the Sparse Principal Component Analysis (SPCA) problem. Most previous research efforts focused on solving the problem of maximizing the variance subject to sparsity constraints or penalizing lack of sparsity. We focus on the problem of minimizing the number of nonzero elements of the loadings subject to a variance constraint. We derive a tractable approach for this problem using Semidefinite Programming (SDP). Our method minimizes a non-convex penalty function mimicking a cardinality penalty function more closely than the convex \(\ell_1\) norm which has been studied before. We develop a novel iterative weighted \(\ell_1\) norm minimization algorithm referred to as PCA Sparsified. We develop two algorithms to solve the weighted \(\ell_1\) norm minimization problem which have different efficiency estimates and computational complexity. Convergence properties of PCA Sparsified are studied. Connections to previously proposed methods are discussed. We introduce a preprocessing method to shrink the problem size which can also be used in previously proposed approaches. Numerical results based on careful implementation show the efficacy and potential of the proposed approach.Keywordssparse PCASDPreweighted optimization \(\ell_1\) norm minimizationCGALADMMMSC codes62H2565K0565K2090C22