Maximizing sum of coupled traces with applications

This paper concerns maximizing the sum of coupled traces of quadratic and linear matrix forms. The coupling comes from requiring the matrix variables in the quadratic and linear matrix forms to be packed together to have orthonormal columns. At a maximum, the KKT condition becomes a nonlinear polar decomposition (NPD) of a matrix-valued function with dependency on the orthogonal polar factor. A self-consistent-field iteration, along with a locally optimal conjugate gradient (LOCG) acceleration, are proposed to compute the NPD. It is proved that both methods are convergent and it is demonstrated numerically that the LOCG acceleration is very effective. As applications, we demonstrate our methods on the MAXBET subproblem and the multi-view partially shared subspace learning (MvPS) subproblem, both of which sit at the computational kernels of two multi-view subspace learning models. In particular, we also demonstrate MvPS on several real world data sets.