Symplectic Accelerated Optimization on SO(3) with Lie Group Variational Integrators

This paper presents computational schemes for optimizing a real-valued function defined on the special orthogonal group. Gradient-based optimization algorithms on a Lie group are interpreted as a continuous-time dynamic system on the group, which is discretized by a Lie group variational integrator that concurrently preserves the symplecticity and the group structure of Hamiltonian systems. It is shown that the proposed approach yields symplectic, accelerated optimization schemes on the special orthogonal group, analogous to classical momentum method. The efficacy of the proposed method is illustrated by numerical examples with an application to spherical shape matching.