Optimization via conformal Hamiltonian systems on manifolds

TL;DR

This paper introduces a novel optimization method on manifolds using conformal Hamiltonian systems, combining symplectic integration and dissipation to improve convergence, demonstrated on sphere minimization tasks.

Contribution

It develops a conformal Hamiltonian framework for manifold optimization, integrating dissipation and symplectic splitting methods, including an adaptive stepsize variant.

Findings

Outperforms gradient descent on sphere minimization

Maintains conformal symplectic structure with constant stepsizes

Effective adaptive stepsize implementation demonstrated

Abstract

In this work we propose a method to perform optimization on manifolds. We assume to have an objective function $f$ defined on a manifold and think of it as the potential energy of a mechanical system. By adding a momentum-dependent kinetic energy we define its Hamiltonian function, which allows us to write the corresponding Hamiltonian system. We make it conformal by introducing a dissipation term: the result is the continuous model of our scheme. We solve it via splitting methods (Lie-Trotter and leapfrog): we combine the RATTLE scheme, approximating the conserved flow, with the exact dissipated flow. The result is a conformal symplectic method for constant stepsizes. We also propose an adaptive stepsize version of it. We test it on an example, the minimization of a function defined on a sphere, and compare it with the usual gradient descent method.