  This paper introduces a new metric and mean on the set of positive
semidefinite matrices of fixed-rank. The proposed metric is derived from a
well-chosen Riemannian quotient geometry that generalizes the reductive
geometry of the positive cone and the associated natural metric. The resulting
Riemannian space has strong geometrical properties: it is geodesically
complete, and the metric is invariant with respect to all transformations that
preserve angles (orthogonal transformations, scalings, and pseudoinversion). A
meaningful approximation of the associated Riemannian distance is proposed,
that can be efficiently numerically computed via a simple algorithm based on
SVD. The induced mean preserves the rank, possesses the most desirable
characteristics of a geometric mean, and is easy to compute.
