Sparse Orthogonal Factor Analysis

We propose a sparse orthogonal factor analysis (SOFA) procedure in which the optimal loadings and unique variances are estimated subject to additional constraint which directly requires some factor loadings to be exact zeros. More precisely, the constraint specifies the required number of zero factor loadings without any restriction on their locations. Such loadings are called sparse which gives the name of the method. SOFA solutions are obtained by minimizing a FA loss function under the sparseness constraint making use of an alternate least squares algorithm. We further present a sparseness selection procedure in which SOFA is performed repeatedly by setting the sparseness at each of a set of feasible integers. Then, the SOFA solution with the optimal sparseness can be chosen using an index for model selection. This procedure allows us to find the optimal orthogonal confirmatory factor analysis model among all possible models. SOFA and the sparseness selection procedure are assessed by simulation and illustrated with well known data sets.