Abstract
We are concerned with the maximization of
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.
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Zhang, L., Li, R. Maximization of the sum of the trace ratio on the Stiefel manifold, I: Theory. Sci. China Math. 57, 2495–2508 (2014). https://doi.org/10.1007/s11425-014-4824-0
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DOI: https://doi.org/10.1007/s11425-014-4824-0
Keywords
- trace ratio
- Rayleigh quotient
- Stiefel manifold
- nonlinear eigenvalue problem
- optimality condition
- eigenspace