Sparsest factor analysis for clustering variables: a matrix decomposition approach

We propose a new procedure for sparse factor analysis (FA) such that each variable loads only onecommon factor. Thus, the loading matrix has a single nonzero element in each rowand zeros elsewhere. Such a loading matrix is the sparsest possible for certain number of variables and common factors. For this reason, the proposed method is named sparsest FA (SSFA). It may also be calledFA-based variable clustering, since the variables loading the same commonfactor can be classified into a cluster.InSSFA, all model parts of FA (common factors, their correlations, loadings, unique factors, and unique variances) are treated as fixed unknown parameter matrices and their least squares function is minimized through specific data matrix decomposition. A useful feature of the algorithm is thatthe matrix of common factor scores isre-parameterized using QR decomposition in order to efficiently estimate factorcorrelations. A simulation study shows that the proposed procedure can exactly identify the true sparsest models. Real data examples demonstrate the usefulness of the variable clustering performed by SSFA.