Maximization of the sum of the trace ratio on the Stiefel manifold, I: Theory

Abstract

We are concerned with the maximization of $$\frac{{tr(V^ \top AV)}} {{tr(V^ \top BV)}} + tr(V^ \top AV) $$ over the Stiefel manifold \(\{ V \in \mathbb{R}^{m \times \ell } |V^ \top V = I_\ell \} (\ell < m)\), where B is a given symmetric and positive definite matrix, A and C are symmetric matrices, and tr(·) is the trace of a square matrix. This is a subspace version of the maximization problem studied in Zhang (2013), which arises from real-world applications in, for example, the downlink of a multi-user MIMO system and the sparse Fisher discriminant analysis in pattern recognition. We establish necessary conditions for both the local and global maximizers and connect the problem with a nonlinear extreme eigenvalue problem. The necessary condition for the global maximizers offers deep insights into the problem, on the one hand, and, on the other hand, naturally leads to a self-consistent-field (SCF) iteration to be presented and analyzed in detail in Part II of this paper.