Double-variable trace maximization for extreme generalized singular quartets of a matrix pair: A geometric method

In this paper, we consider the problem of computing an arbitrary generalized singular value of a Grassman or real matrix pair and a triplet of associated generalized singular vectors. Based on the QR factorization, the problem is reformulated as two novel trace maximization problems, each of which has double variables with unitary constraints or orthogonal constraints. Theoretically, we show that the arbitrarily prescribed extreme generalized singular values and associated triplets of generalized singular vectors can be determined by the global solutions of the constrained trace optimization problems. Then we propose a geometric inexact Newton–conjugate gradient (Newton-CG) method for solving their equivalent trace minimization problems over the Riemannian manifold of all fixed-rank partial isometries. The proposed method can extract not only the prescribed extreme generalized singular values but also associated triplets of generalized singular vectors. Under some mild assumptions, we establish the global and quadratic convergence of the proposed method. Finally, numerical experiments on both synthetic and real data sets show the effectiveness and high accuracy of our method.