Maximum Likelihood Covariance Estimation with a Condition Number Constraint

In many signal processing applications, we want to estimate the covariance matrix of a multivariate Gaussian distribution. We often require the estimate to be not only invertible but also well-conditioned. We consider the maximum likelihood estimation of the covariance matrix with a constraint on the condition number. We show that this estimation problem can be reformulated as a convex univariate minimization problem, which admits an analytic solution. This estimation method requires no special assumption on the structure of the true covariance matrix. We demonstrate its good performance in comparison with commonly used estimators, especially when the sample size is small.