Manifolds, tensor analysis, and applications: 2nd edition
Manifolds, tensor analysis, and applications: 2nd edition
An Optimal Transformation for Discriminant and Principal Component Analysis
IEEE Transactions on Pattern Analysis and Machine Intelligence
The projected gradient methods for least squares matrix approximations with spectral constraints
SIAM Journal on Numerical Analysis
Introduction to statistical pattern recognition (2nd ed.)
Introduction to statistical pattern recognition (2nd ed.)
Deflation Techniques for an Implicitly Restarted Arnoldi Iteration
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
The symmetric eigenvalue problem
The symmetric eigenvalue problem
The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
Trust-region methods
SIAM Journal on Matrix Analysis and Applications
Pattern Recognition and Machine Learning (Information Science and Statistics)
Pattern Recognition and Machine Learning (Information Science and Statistics)
Trust-Region Methods on Riemannian Manifolds
Foundations of Computational Mathematics
An Optimal Set of Discriminant Vectors
IEEE Transactions on Computers
Optimization Algorithms on Matrix Manifolds
Optimization Algorithms on Matrix Manifolds
Accelerated Line-search and Trust-region Methods
SIAM Journal on Numerical Analysis
A truncated-CG style method for symmetric generalized eigenvalue problems
Journal of Computational and Applied Mathematics
Uncorrelated trace ratio linear discriminant analysis for undersampled problems
Pattern Recognition Letters
Fast Algorithms for the Generalized Foley-Sammon Discriminant Analysis
SIAM Journal on Matrix Analysis and Applications
An optimization criterion for generalized discriminant analysis on undersampled problems
IEEE Transactions on Pattern Analysis and Machine Intelligence
On a self-consistent-field-like iteration for maximizing the sum of the Rayleigh quotients
Journal of Computational and Applied Mathematics
| Hi-index | 0.00 |
Given symmetric matrices B,D驴驴 n脳n and a symmetric positive definite matrix W驴驴 n脳n , maximizing the sum of the Rayleigh quotient x 驴 D x and the generalized Rayleigh quotient $\frac{\mathbf{x}^{\top}B \mathbf{x}}{\vphantom{\mathrm{I}^{\mathrm{I}}}\mathbf{x}^{\top}W\mathbf{x}}$ on the unit sphere not only is of mathematical interest in its own right, but also finds applications in practice. In this paper, we first present a real world application arising from the sparse Fisher discriminant analysis. To tackle this problem, our first effort is to characterize the local and global maxima by investigating the optimality conditions. Our results reveal that finding the global solution is closely related with a special extreme nonlinear eigenvalue problem, and in the special case D=μW (μ0), the set of the global solutions is essentially an eigenspace corresponding to the largest eigenvalue of a specially-defined matrix. The characterization of the global solution not only sheds some lights on the maximization problem, but motives a starting point strategy to obtain the global maximizer for any monotonically convergent iteration. Our second part then realizes the Riemannian trust-region method of Absil, Baker and Gallivan (Found. Comput. Math. 7:303---330, 2007) into a practical algorithm to solve this problem, which enjoys the nice convergence properties: global convergence and local superlinear convergence. Preliminary numerical tests are carried out and empirical evaluation of its performance is reported.