Chordal Metric Formula Between Generalized Singular Values of Grassmann Matrix Pairs by Riemannian Optimization Models

In this paper, we present a new explicit formula for the sum of the chordal distance between the generalized singular values of Grassmann matrix pairs, based on Riemannian optimization models. The new formulas involve small-size unitary matrices and real orthogonal matrices derived from Riemannian optimization models. We then apply Newton’s method on Riemannian manifolds to efficiently solve the variable matrices involved. The new explicit formulas provide significant improvements over the existing theoretical and computational upper bounds.