On Estimation of Sparse Factor Loadings Using Distribution-free Approach

Abstract

Sparse Factor Analysis (SFA) is often used for the analysis of high dimensional data, providing simpler pattern of factor loadings by constraining insignificant loadings to be zero. However, existing SFA approaches require the assumption of normality of data since sparse factor loadings are obtained through a likelihood function with additional constraint or penalty function. This work proposes a method for obtaining sparse factor loadings without requiring any distributional assumption. In this method, the orthogonal sparse eigenvectors were computed based on Procrustes reformulation, and thereafter, an iterative procedure was provided to find sparse factor loadings corresponding to the orthogonal sparse eigenvectors. In the end, the proposed method was compared with penalized likelihood factor analysis via nonconvex penalties using simulated data. Results show that sparse factor loadings from both methods provide simpler structure of factor loadings than the structure obtained from standard Exploratory Factor Analysis. In addition, the new method out-performs the penalized likelihood factor analysis via nonconvex penalties as it provides smaller values of MSE even when the two methods have the same level of sparsity.