基于正则化秩k矩阵逼近的稀疏主成分分析

发布时间：2018-05-14 03:03

本文选题：降维 + 稀疏主成分　；参考：《北京航空航天大学学报》2017年06期

【摘要】：在计算稀疏主成分(PCs)时,由于同时求k个主成分的做法可以减少计算所产生的累积误差,因此提出了基于正则化秩k矩阵逼近的稀疏主成分模型,并设计了求解该模型的块坐标下降法(BCD-s PCA-r SVD)。该算法的主要思想是先把变量按坐标分成2k个块,当固定其他2k-1个坐标块的变量时,求解关于单个坐标块的子问题并给出子问题的显式解,循环地求解这些子问题直至满足终止条件。该算法每次迭代的计算复杂度关于样本个数与变量维数都是线性的,并且证明了它是收敛的。该算法不仅易于实现,数值仿真结果表明,该算法应用到真实数据与合成数据上都是可行且有效的。它不仅使累积误差降低,而且具有较低的计算复杂度,因而可以有效地求解大规模稀疏主成分分析问题。
[Abstract]:In the calculation of sparse principal component (PCS), a sparse principal component model based on regularized rank k matrix approximation is proposed because the cumulative error can be reduced by finding k principal components at the same time. The block coordinate descent method is designed to solve the model. The main idea of the algorithm is to divide the variables into 2k blocks according to the coordinates. When the variables of other 2k-1 coordinate blocks are fixed, the subproblems of the single coordinate blocks are solved and the explicit solutions of the sub-problems are given. These subproblems are solved circularly until the termination conditions are satisfied. The computational complexity of the algorithm for each iteration is linear with respect to the number of samples and the dimension of variables, and it is proved that the algorithm is convergent. The numerical simulation results show that the algorithm is feasible and effective when applied to both real and composite data. It not only reduces the cumulative error, but also has low computational complexity, so it can effectively solve the large scale sparse principal component analysis problem.
【作者单位】：北京航空航天大学数学与系统科学学院;
【基金】：国家自然科学基金(61172060,61403011)~~
【分类号】：TP301.6

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