Accelerated Optimization on Riemannian Manifolds via Discrete Constrained Variational Integrators

Area of Science:

• Mathematics
• Numerical Analysis
• Optimization

Background:

• A variational formulation for accelerated optimization was established for normed vector spaces and generalized to Riemannian manifolds.
• Previous work demonstrated the efficiency of time-adaptive geometric integrators for symplectic accelerated optimization on normed vector spaces, highlighting their robustness and computational advantages.
• Geometric discretizations preserving time-rescaling invariance and symplecticity proved superior in stability and efficiency.

Purpose of the Study:

• To develop time-adaptive Hamiltonian variational integrators for accelerated optimization on Riemannian manifolds.
• To explore the incorporation of holonomic constraints within discrete variational integrators for Riemannian Hamiltonian systems.
• To test the performance of these algorithms on eigenvalue and Procrustes problems on the unit sphere and Stiefel manifold.

Main Methods:

• Developing time-adaptive Hamiltonian variational integrators tailored for Riemannian manifolds.
• Incorporating holonomic constraints into discrete variational integrators to ensure numerical discretization stays on the manifold.
• Applying the developed integrators to solve optimization problems such as eigenvalue and Procrustes problems.

Main Results:

• The proposed integrators are designed to be robust and computationally efficient for accelerated optimization on Riemannian manifolds.
• Holonomic constraints are effectively used to maintain numerical solutions on the manifold.
• Demonstrated performance on benchmark problems like eigenvalue and Procrustes problems.

Conclusions:

• Time-adaptive Hamiltonian variational integrators offer a promising approach for accelerated optimization on Riemannian manifolds.
• The integration of holonomic constraints is crucial for the accurate and stable numerical discretization of Riemannian Hamiltonian systems.
• The developed algorithms show potential for solving challenging optimization problems in various scientific domains.