  This paper considers the problem of robustly estimating a structured
covariance matrix with an elliptical underlying distribution with known mean.
In applications where the covariance matrix naturally possesses a certain
structure, taking the prior structure information into account in the
estimation procedure is beneficial to improve the estimation accuracy. We
propose incorporating the prior structure information into Tyler's M-estimator
and formulate the problem as minimizing the cost function of Tyler's estimator
under the prior structural constraint. First, the estimation under a general
convex structural constraint is introduced with an efficient algorithm for
finding the estimator derived based on the majorization minimization (MM)
algorithm framework. Then, the algorithm is tailored to several special
structures that enjoy a wide range of applications in signal processing related
fields, namely, sum of rank-one matrices, Toeplitz, and banded Toeplitz
structure. In addition, two types of non-convex structures, i.e., the Kronecker
structure and the spiked covariance structure, are also discussed, where it is
shown that simple algorithms can be derived under the guidelines of MM.
Numerical results show that the proposed estimator achieves a smaller
estimation error than the benchmark estimators at a lower computational cost.
