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.

翻译：相较于欧几里得空间，双曲空间在处理具有内在层次结构的数据方面日益展现出卓越的性能。然而，在双曲空间中进行学习面临着重大挑战。具体而言，将支持向量机扩展至双曲空间通常是一个约束非凸优化问题。先前解决双曲支持向量机的流行方法主要采用投影梯度下降法，这类方法通常对超参数和初始化条件敏感，常导致次优解。本工作中，我们首先将问题重写为多项式优化形式，进而应用半定松弛与稀疏矩-平方和松弛技术来有效逼近最优解。大量实证实验表明，这些方法的性能优于投影梯度下降法。