• Corpus ID: 232478310

On Generalizing Trace Minimization

@article{Liang2021OnGT,
  title={On Generalizing Trace Minimization},
  author={Xin Liang and Li Wang and Lei-Hong Zhang and Ren-Cang Li},
  journal={ArXiv},
  year={2021},
  volume={abs/2104.00257}
}
Ky Fan’s trace minimization principle is extended along the line of the Brockett cost function tr(DXAX) in X on the Stiefel manifold, where D of an apt size is positive definite. Specifically, we investigate infX tr(DX AX) subject to XBX = Ik or Jk = diag(±1). We establish conditions under which the infimum is finite and when it is finite, analytic solutions are obtained in terms of the eigenvalues and eigenvectors of the matrix pencil A−λB, where B is possibly indefinite and singular, and D is… 
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References

SHOWING 1-10 OF 31 REFERENCES
Trace minimization and definiteness of symmetric pencils
Trace minimization principles for positive semi-definite pencils
Quadratic optimization with orthogonality constraint: explicit Łojasiewicz exponent and linear convergence of retraction-based line-search and stochastic variance-reduced gradient methods
TLDR
The growth behavior of the objective function around the critical points of the QP-OC problem is characterized and it is demonstrated how such characterization can be used to obtain strong convergence rate results for iterative methods that exploit the manifold structure of the orthogonality constraint to find a critical point of the problem.
First order optimality conditions and steepest descent algorithm on orthogonal Stiefel manifolds
TLDR
The steepest descent algorithm on the orthogonal Stiefel manifold is described using the ambient coordinates and not the local coordinates of the manifold, and the dependence of the recurrence sequence that defines the algorithm depends on the choice of a full rank submatrix.
Maximization of the sum of the trace ratio on the Stiefel manifold, I: Theory
TLDR
The necessary conditions for both the local and global maximizers are established and a self-consistent-field (SCF) iteration is presented and analyzed in detail in Part II of this paper.
A Jacobi eigenreduction algorithm for definite matrix pairs
SummaryWe propose a Jacobi eigenreduction algorithm for symmetric definite matrix pairsA, J of small to medium-size real symmetric matrices withJ2=I,J diagonal (neitherJ norA itself need be…
Trace optimization and eigenproblems in dimension reduction methods
TLDR
All the eigenvalue problems solved in the context of explicit linear projections can be viewed as the projected analogues of the nonlinear or implicit projections, including kernels as a means of unifying linear and nonlinear methods.
Maximization of the sum of the trace ratio on the Stiefel manifold, II: Computation
TLDR
This part analyzes the global and local convergence of the SCF iteration, and shows that the necessary condition for the global maximizers is fulfilled at any convergent point of the sequences of approximations generated by theSCF iteration.
The Trace Ratio Optimization Problem for Dimensionality Reduction
TLDR
The goal of this paper is to examine this trace ratio optimization problem in detail, to consider different algorithms for solving it, and to illustrate the use of these algorithms for dimensionality reduction.
Rayleigh Quotient Based Optimization Methods For Eigenvalue Problems
Four classes of eigenvalue problems that admit similar min-max principles and the Cauchy interlacing inequalities as the symmetric eigenvalue problem famously does are investigated. These min-max…
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