  Factor Analysis (FA) is a technique of fundamental importance that is widely
used in classical and modern multivariate statistics, psychometrics and
econometrics. In this paper, we revisit the classical rank-constrained FA
problem, which seeks to approximate an observed covariance matrix
($\boldsymbol\Sigma$), by the sum of a Positive Semidefinite (PSD) low-rank
component ($\boldsymbol\Theta$) and a diagonal matrix ($\boldsymbol\Phi$) (with
nonnegative entries) subject to $\boldsymbol\Sigma - \boldsymbol\Phi$ being
PSD. We propose a flexible family of rank-constrained, nonlinear Semidefinite
Optimization based formulations for this task. We introduce a reformulation of
the problem as a smooth optimization problem with convex compact constraints
and propose a unified algorithmic framework, utilizing state of the art
techniques in nonlinear optimization to obtain high-quality feasible solutions
for our proposed formulation. At the same time, by using a variety of
techniques from discrete and global optimization, we show that these solutions
are certifiably optimal in many cases, even for problems with thousands of
variables. Our techniques are general and make no assumption on the underlying
problem data. The estimator proposed herein, aids statistical interpretability,
provides computational scalability and significantly improved accuracy when
compared to current, publicly available popular methods for rank-constrained
FA. We demonstrate the effectiveness of our proposal on an array of synthetic
and real-life datasets. To our knowledge, this is the first paper that
demonstrates how a previously intractable rank-constrained optimization problem
can be solved to provable optimality by coupling developments in convex
analysis and in discrete optimization.
