Factor analysis models via I-divergence optimization

Given a positive definite covariance matrix Σˆ of dimension n, we approximate it with a covariance of the form HH⊤+D, where H has a prescribed number k<n of columns and D>0 is diagonal. The quality of the approximation is gauged by the I-divergence between the zero mean normal laws with covariances Σˆ and HH⊤+D, respectively. To determine a pair (H, D) that minimizes the I-divergence we construct, by lifting the minimization into a larger space, an iterative alternating minimization algorithm (AML) à la Csiszár-Tusnády. As it turns out, the proper choice of the enlarged space is crucial for optimization. The convergence of the algorithm is studied, with special attention given to the case where D is singular. The theoretical properties of the AML are compared to those of the popular EM algorithm for exploratory factor analysis. Inspired by the ECME (a Newton-Raphson variation on EM), we develop a similar variant of AML, called ACML, and in a few numerical experiments, we compare the performances of the four algorithms.