  This paper proposes new algorithms for the metric learning problem. We start
by noticing that several classical metric learning formulations from the
literature can be viewed as modified covariance matrix estimation problems.
Leveraging this point of view, a general approach, called Robust Geometric
Metric Learning (RGML), is then studied. This method aims at simultaneously
estimating the covariance matrix of each class while shrinking them towards
their (unknown) barycenter. We focus on two specific costs functions: one
associated with the Gaussian likelihood (RGML Gaussian), and one with Tyler's M
-estimator (RGML Tyler). In both, the barycenter is defined with the Riemannian
distance, which enjoys nice properties of geodesic convexity and affine
invariance. The optimization is performed using the Riemannian geometry of
symmetric positive definite matrices and its submanifold of unit determinant.
Finally, the performance of RGML is asserted on real datasets. Strong
performance is exhibited while being robust to mislabeled data.
