  In exploratory factor analysis, model parameters are usually estimated by
maximum likelihood method. The maximum likelihood estimate is obtained by
solving a complicated multivariate algebraic equation. Since the solution to
the equation is usually intractable, it is typically computed with continuous
optimization methods, such as Newton-Raphson methods. With this procedure,
however, the solution is inevitably dependent on the estimation algorithm and
initial value since the log-likelihood function is highly non-concave.
Particularly, the estimates of unique variances can result in zero or negative,
referred to as improper solutions; in this case, the maximum likelihood
estimate can be severely unstable. To delve into the issue of the instability
of the maximum likelihood estimate, we compute exact solutions to the
multivariate algebraic equation by using algebraic computations. We provide a
computationally efficient algorithm based on the algebraic computations
specifically optimized for maximum likelihood factor analysis. To be specific,
Gr\"oebner basis and cylindrical decomposition are employed, powerful tools for
solving the multivariate algebraic equation. Our proposed procedure produces
all exact solutions to the algebraic equation; therefore, these solutions are
independent of the initial value and estimation algorithm. We conduct Monte
Carlo simulations to investigate the characteristics of the maximum likelihood
solutions.
