Optimization on the Surface of the (Hyper)-Sphere
This work addresses a computationally intensive physics optimization problem, but it is incremental as it applies existing methods without major breakthroughs.
The authors tackled the Thomson problem of optimizing charged particle distributions on high-dimensional spheres, finding that stochastic gradient descent (SGD) scales well with the number of points in numerical experiments.
Thomson problem is a classical problem in physics to study how $n$ number of charged particles distribute themselves on the surface of a sphere of $k$ dimensions. When $k=2$, i.e. a 2-sphere (a circle), the particles appear at equally spaced points. Such a configuration can be computed analytically. However, for higher dimensions such as $k \ge 3$, i.e. the case of 3-sphere (standard sphere), there is not much that is understood analytically. Finding global minimum of the problem under these settings is particularly tough since the optimization problem becomes increasingly computationally intensive with larger values of $k$ and $n$. In this work, we explore a wide variety of numerical optimization methods to solve the Thomson problem. In our empirical study, we find stochastic gradient based methods (SGD) to be a compelling choice for this problem as it scales well with the number of points.