From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices

Here, we prove Theorem 1 from Section 3, i.e., the equivalence between the length of any given curve under the geodesic distance δg and the Stein metric δS up to scale of 2 √ 2. The proof of this theorem follows several steps. We start with the definition of curve length and intrinsic metric. Without any assumption on differentiability, let (M, d) be a metric space. A curve inM is a continuous function γ : [0, 1] →M and joins the starting point γ(0) = x to the end point γ(1) = y.