Approximate factor analysis model building via alternating I-divergence minimization

Given a positive definite covariance matrix $\widehat \Sigma$, we strive to construct an optimal \emph{approximate} factor analysis model $HH^\top +D$, with $H$ having a prescribed number of columns and $D>0$ diagonal. The optimality criterion we minimize is the I-divergence between the corresponding normal laws. Lifting the problem into a properly chosen larger space enables us to derive an alternating minimization algorithm \`a la Csisz\'ar-Tusn\'ady for the construction of the best approximation. The convergence properties of the algorithm are studied, with special attention given to the case where $D$ is singular.