联合L1范数和迹范数的数据降维模型及其优化算法

Dimension Reduction Model via Joint L1-trace Norms and Optimization Algorithm

主成分分析(PCA)是多元分析中广泛应用的降维方法,但是传统的降维模型一般是基于矩阵的秩,然而秩的计算是非凸、不连续的问题且计算复杂。本文针对这一问题,提出联合更具鲁棒性的L1范数和具有凸性的迹范数建立一种联合数据降维模型,针对模型的优化提出基于拉格朗日乘子的优化算法。最后将模型应用于UCI数据集以及Yale人脸数据集和扩展Yale B人脸数据集进行数据处理。数学分析和可视化实验结果都表明模型和优化算法是有效的。

Principal component analysis （PCA） is widely used in multivariate analysis for dimension reduction. Traditional data dimension reduction is generally based on the rank of the matrix, and the rank calculation is non-convex and discrete issues is complex. In order to solve this problem, the paper proposed the model based on a robust formulation using L1 norm together with trace norm. The paper derived an efficient ALM algorithm for nonlinear optimization and applied it in UCI data sets and some image data sets against noises. Both mathematical analysis and visual results show the efficiency and good performance of the proposed method.