Least-Squares on the Real Symplectic Group

TL;DR

This paper develops a gradient-descent algorithm on the pseudo-Riemannian manifold of the real symplectic group to solve least-squares problems, enabling efficient optimization on this curved space.

Contribution

It introduces a geodesic-based gradient descent method tailored for the non-compact Lie group Sp(2n,R) using pseudo-Riemannian geometry.

Findings

Closed-form geodesic distance computation on Sp(2n,R)

Effective gradient descent algorithm for least-squares on symplectic group

Extension of manifold optimization techniques to non-compact Lie groups

Abstract

The present paper discusses the problem of least-squares over the real symplectic group of matrices Sp(2n,R)$. The least-squares problem may be extended from flat spaces to curved spaces by the notion of geodesic distance. The resulting non-linear minimization problem on manifold may be tackled by means of a gradient-descent algorithm tailored to the geometry of the space at hand. In turn, gradient steepest descent on manifold may be implemented through a geodesic-based stepping method. As the space Sp(2n,R) is a non-compact Lie group, it is convenient to endow it with a pseudo-Riemannian geometry. Indeed, a pseudo-Riemannian metric allows the computation of geodesic arcs and geodesic distances in closed form on Sp(2n,R).