# Optimization on the symplectic group

arXiv: https://arxiv.org/abs/1811.07345 · HTML: https://mlsanity.com/paper/1811.07345/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Nov 2018.

## Built on by (most cited first, 1 of 1)

- [MetaSym: A Symplectic Meta-learning Framework for Physical Intelligence](https://mlsanity.com/paper/2502.16667/) (2025)

## Abstract (arXiv)

We regard the real symplectic group Sp(2n,ℝ) as a constraint submanifold of the 2n× 2n real matrices M2n(ℝ) endowed with the Euclidean (Frobenius) metric, respectively as a submanifold of the general linear group Gl(2n,ℝ) endowed with the (left) invariant metric. For a cost function that defines an optimization problem on the real symplectic group we give a necessary and sufficient condition for critical points and we apply this condition to the particular case of a least square cost function. In order to characterize the critical points we give a formula for the Hessian of a cost function defined on the real symplectic group, with respect to both considered metrics. For a generalized Brockett cost function we present a necessary condition and a sufficient condition for local minimum. We construct a retraction map that allows us to detail the steepest descent and embedded Newton algorithms for solving an optimization problem on the real symplectic group.