On-line Kronecker Product Structured Covariance Estimation with Riemannian geometry for t-distributed data

The information geometry of the zero-mean t-distribution with Kronecker-product structured covariance matrix is derived. In particular, we obtain the Fisher information metric which shows that this geometry is identifiable to a product manifold of
$\mathcal{S}_{p}^{++}$
(positive definite symmetric matrices) and
$s\mathcal{S}_{p}^{++}$
(positive definite symmetric matrices with unit determinant). From this result, we obtain the geodesics and the Riemannian gradient. Finally, this geometry makes it possible to propose an on-line covariance matrix estimation algorithm well adapted to large datasets. Numerical experiments show that optimal results are obtained for a reasonable number of data.