Some results on constrained maximum likelihood estimation

This paper considers, for a multivariate Gaussian random process, the maximum likelihood estimation (MLE) of a covariance matrix whose structure satisfies some particular constraints. First, one examines the case where the random process is required to satisfy a time varying auto-regressive (AR) model of fixed order p. In particular, one shows that the resulting optimal covariance matrix is a partial reconstruction of the given sample covariance matrix. Next, a linear feature extraction is considered with a slightly unusual criterion which requires that the likelihood of the extracted features should be as large as possible.