  Hyperbolic spaces have increasingly been recognized for their outstanding
performance in handling data with inherent hierarchical structures compared to
their Euclidean counterparts. However, learning in hyperbolic spaces poses
significant challenges. In particular, extending support vector machines to
hyperbolic spaces is in general a constrained non-convex optimization problem.
Previous and popular attempts to solve hyperbolic SVMs, primarily using
projected gradient descent, are generally sensitive to hyperparameters and
initializations, often leading to suboptimal solutions. In this work, by first
rewriting the problem into a polynomial optimization, we apply semidefinite
relaxation and sparse moment-sum-of-squares relaxation to effectively
approximate the optima. From extensive empirical experiments, these methods are
shown to perform better than the projected gradient descent approach.
