Abstract
In this paper, we describe a novel approach to sparse principal component analysis (SPCA) via a nonconvex sparsity-inducing fraction penalty function SPCA (FP-SPCA). Firstly, SPCA is reformulated as a fraction penalty regression problem model. Secondly, an algorithm corresponding to the model is proposed and the convergence of the algorithm is guaranteed. Finally, numerical experiments were carried out on a synthetic data set, and the experimental results show that the FP-SPCA method is more adaptable and has a better performance in the tradeoff between sparsity and explainable variance than SPCA.
Highlights
Principal component analysis (PCA) [1] has become more popular in data analysis, dimension reduction, image processing, and feature extraction [2]
En, the columns of matrix U D are the principal components of X, and the columns of V are the corresponding loadings of the principal components. e principle component analysis (PCA) method is popular due to the following two properties: first, principal components capture the maximum variability from the columns of matrix X, which guarantees minimal information loss; second, principal components are uncorrelated, so we can use one of them without considering the others
Jollife et al [18] first proposed a sparse principal component analysis (SPCA) method using l1-norm regularization which leads to a variety of sparsity-inducing methods. e success of SPCA in gaining more interpretable model motivates us to propose the method in this paper, where we proposed a fractional function penalty method with respect to sparse PCA. e method is to exploit the fractional function [19] to penalize the model coefficients in order to obtain sparse coefficients, which makes the PCA to have more interpretable ability
Summary
Principal component analysis (PCA) [1] has become more popular in data analysis, dimension reduction, image processing, and feature extraction [2]. E PCA method is popular due to the following two properties: first, principal components capture the maximum variability from the columns of matrix X, which guarantees minimal information loss; second, principal components are uncorrelated, so we can use one of them without considering the others. PCA has its own drawback, i.e., each principal component of matrix X is a linear combination of all p variables and the elements of the loading vectors (the columns of matrix V) are usually nonzero, which make them difficult to interpret the gained PCs. Based on the abovementioned priority properties and drawback, some scholars consider it will be a wise option to keep the dimension reduction property at the same time to reduce the number of explainable variables. Without specification, scalar is denoted as lower case letter x, and vector is denoted as bold lower case letter x (x1, x2, . . . , xn)⊤. e matrix is denoted by bold capital letter M, and I denotes the identity matrix. e transpose of a real matrix M is denoted by M⊤. e Frobenius norm of · is denoted by ‖ · ‖, where · denotes column vector or matrix, and the spectral norm of a matrix M is denoted by ‖M‖2
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