Extended Hamiltonian learning on Riemannian manifolds: numerical aspects

Area of Science:

• Computational mathematics
• Machine learning theory
• Differential geometry

Background:

• Extends prior theoretical work on learning on manifolds using Hamiltonian principles.
• Addresses the need for numerical methods in manifold-based learning frameworks.

Purpose of the Study:

• To present the numerical implementation of the extended Hamiltonian learning paradigm.
• To demonstrate the application of geometric numerical integration for solving learning equations on manifolds.

Main Methods:

• Utilizes geometric numerical integration techniques.
• Applies general-purpose integration schemes to dynamical learning equations.
• Employs numerical examples and case studies for validation.

Main Results:

• The numerical implementation of the extended Hamiltonian learning paradigm is detailed.
• The dynamical learning equations exhibit a rich and structured behavior.
• Numerical examples validate the theoretical framework presented in both parts of the study.

Conclusions:

• The numerical approach confirms the viability and effectiveness of the extended Hamiltonian learning paradigm.
• Geometric numerical integration provides a robust framework for implementing manifold-based learning.
• The study bridges theoretical concepts with practical numerical solutions in machine learning on manifolds.