Geometric distance and mean for positive semi-definite matrices of fixed rank

This paper introduces a new distance and mean on the set of positive semi-definite matrices of fixed-rank. The proposed distance 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, dilations, and pseudo-inversion). The associated distance 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.