Critical Topology for Optimization on the Symplectic Group

TL;DR

This paper investigates the critical topology of least squares optimization problems on the noncompact symplectic group, revealing a complex structure with a unique minimum and saddle points, and demonstrating the absence of traps for optimization.

Contribution

It provides the first detailed analysis of the critical topology for optimization on the noncompact symplectic group, highlighting differences from compact Lie groups.

Findings

Unique local minimum identified

Presence of saddle points in the topology

No traps ensure global convergence of algorithms

Abstract

Optimization problems over compact Lie groups have been extensively studied due to their broad applications in linear programming and optimal control. This paper analyzes least square problems over a noncompact Lie group, the symplectic group $\Sp(2N,\R)$, which can be used to assess the optimality of control over dynamical transformations in classical mechanics and quantum optics. The critical topology for minimizing the Frobenius distance from a target symplectic transformation is solved. It is shown that the critical points include a unique local minimum and a number of saddle points. The topology is more complicated than those of previously studied problems on compact Lie groups such as the orthogonal and unitary groups because the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group brings significant nonlinearity to the problem. Nonetheless, the lack of traps guarantees the global convergence of local optimization algorithms.