Penalized maximum-likelihood estimation of covariance matrices with linear structure

y In this paper, a space-alternating generalized expectation-maximization (SAGE) algorithm is presented for the numerical computation of maximum-likelihood (ML) and penalized maximum-likelihood (PML) estimates of the parameters of covariance matrices with linear structure for complex Gaussian processes. By using a less informative hidden-data space and a sequential parameter-update scheme, a SAGE-based algorithm is derived for which convergence of the likelihood is demonstrated to be signiicantly faster than that of an EM-based algorithm that has been previously proposed. In addition, the SAGE procedure is shown to easily accommodate penalty functions, and a SAGE-based algorithm is derived and demonstrated for forming PML estimates with a quadratic smoothness penalty. 3 Maximum-likelihood estimates obtained by applying 500 iterations of the EM-and SAGE-based algorithms: true parameters (dashed); EM or SAGE estimate (solid). : 10 4 Comparison of modiied log-likelihood versus iterations for the EM-and SAGE-based 5 Conditional Fisher information as a function of for various noise-to-signal ratios. 6 Maximum-likelihood estimates obtained by applying 100 iterations of the EM-and SAGE-based algorithms: true parameters (dashed); EM or SAGE estimate (solid). : 17 7 Comparison of modiied log-likelihood versus iterations for the EM-and SAGE-based 8 Maximum-likelihood parameter estimates obtained by applying 10 and 20 iterations of SAGE-based algorithm with = 1: true parameters (dashed); SAGE estimate 9 PML estimates obtained by applying the SAGE-based algorithm for various values of : true parameters (dashed); SAGE estimates (solid).