  Factor analysis, a classical multivariate statistical technique is popularly
used as a fundamental tool for dimensionality reduction in statistics,
econometrics and data science. Estimation is often carried out via the Maximum
Likelihood (ML) principle, which seeks to maximize the likelihood under the
assumption that the positive definite covariance matrix can be decomposed as
the sum of a low rank positive semidefinite matrix and a diagonal matrix with
nonnegative entries. This leads to a challenging rank constrained nonconvex
optimization problem. We reformulate the low rank ML Factor Analysis problem as
a nonlinear nonsmooth semidefinite optimization problem, study various
structural properties of this reformulation and propose fast and scalable
algorithms based on difference of convex (DC) optimization. Our approach has
computational guarantees, gracefully scales to large problems, is applicable to
situations where the sample covariance matrix is rank deficient and adapts to
variants of the ML problem with additional constraints on the problem
parameters. Our numerical experiments demonstrate the significant usefulness of
our approach over existing state-of-the-art approaches.
