114.
Given a positive definite covariance matrix
\(\widehat{\Sigma }\) of dimension
n, we approximate it with a covariance of the form
\(HH^\top +D\), 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
\(\widehat{\Sigma }\) and
\(HH^\top +D\), 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.
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