  The Riemannian geometry of covariance matrices has been essential to several
successful applications, in computer vision, biomedical signal and image
processing, and radar data processing. For these applications, an important
ongoing challenge is to develop Riemannian-geometric tools which are adapted to
structured covariance matrices. The present paper proposes to meet this
challenge by introducing a new class of probability distributions, Gaussian
distributions of structured covariance matrices. These are Riemannian analogs
of Gaussian distributions, which only sample from covariance matrices having a
preassigned structure, such as complex, Toeplitz, or block-Toeplitz. The
usefulness of these distributions stems from three features: (1) they are
completely tractable, analytically or numerically, when dealing with large
covariance matrices, (2) they provide a statistical foundation to the concept
of structured Riemannian barycentre (i.e. Fr\'echet or geometric mean), (3)
they lead to efficient statistical learning algorithms, which realise, among
others, density estimation and classification of structured covariance
matrices. The paper starts from the observation that several spaces of
structured covariance matrices, considered from a geometric point of view, are
Riemannian symmetric spaces. Accordingly, it develops an original theory of
Gaussian distributions on Riemannian symmetric spaces, of their statistical
inference, and of their relationship to the concept of Riemannian barycentre.
Then, it uses this original theory to give a detailed description of Gaussian
distributions of three kinds of structured covariance matrices, complex,
Toeplitz, and block-Toeplitz. Finally, it describes algorithms for density
estimation and classification of structured covariance matrices, based on
Gaussian distribution mixture models.
