Solving Minimal-Distance Problems over the Manifold of Real-Symplectic Matrices

@article{Fiori2011SolvingMP,
  title={Solving Minimal-Distance Problems over the Manifold of Real-Symplectic Matrices},
  author={Simone G. O. Fiori},
  journal={SIAM J. Matrix Anal. Appl.},
  year={2011},
  volume={32},
  pages={938-968}
}
  • S. Fiori
  • Published 20 September 2011
  • Mathematics
  • SIAM J. Matrix Anal. Appl.
The present paper discusses the question of formulating and solving minimal-distance problems over the group-manifold of real-symplectic matrices. In order to tackle the related optimization problem, the real-symplectic group is regarded as a pseudo-Riemannian manifold, and a metric is chosen that affords the computation of geodesic arcs in closed forms. Then, the considered minimal-distance problem can be solved numerically via a gradient-steepest-descent algorithm implemented through a… 

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References

SHOWING 1-10 OF 43 REFERENCES

Critical Landscape Topology for Optimization on the Symplectic Group

Optimization problems over compact Lie groups have been studied extensively due to their broad applications in linear programming and optimal control. This paper analyzes an optimization problem over

Parametrization of the Matrix Symplectic Group and Applications

The group of symplectic matrices is explicitly parametrized, and this description is applied to solve two types of problems to describe those matrices that can be certain significant submatrices of a symplectic matrix, and to parametrization of the symp eclectic matrices with a given matrix occurring as a submatrix in a given position.

Optimal Control and Geodesics on Quadratic Matrix Lie Groups

A discretization of this symmetric representation is constructed making use of the symmetry, which in turn give rise to numerical algorithms to integrate the generalized rigid body equations for the given class of matrix Lie groups.

Learning by Natural Gradient on Noncompact Matrix-Type Pseudo-Riemannian Manifolds

  • S. Fiori
  • Mathematics
    IEEE Transactions on Neural Networks
  • 2010
This paper deals with learning by natural-gradient optimization on noncompact manifolds endowed with pseudo-Riemannian metrics, which may give rise to tractable calculations.

Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices

This work defines the Log‐Euclidean mean from a Riemannian point of view, based on a lie group structure which is compatible with the usual algebraic properties of this matrix space and a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure.

Minimizing a differentiable function over a differential manifold

To generalize the descent methods of unconstrained optimization to the constrained case, we define intrinsically the gradient field of the objective function on the constraint manifold and analyze

The Gradient Projection Method Along Geodesics

The method of steepest descent for solving unconstrained minimization problems is well understood. It is known, for instance, that when applied to a smooth objective function f, and converging to a

Numerical methods for ordinary differential equations on matrix manifolds

Optimization Algorithms on Matrix Manifolds

Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis and will be of interest to applied mathematicians, engineers, and computer scientists.