We study the problem of constructing Steiner Minimal Trees (SMTs) in hyperbolic space. Exact SMT computation is NP-hard, and existing hyperbolic heuristics such as HyperSteiner are deterministic and often get trapped in locally suboptimal configurations. We introduce Randomized HyperSteiner (RHS), a stochastic Delaunay triangulation heuristic that incorporates randomness into the expansion process and refines candidate trees via Riemannian gradient descent optimization. Experiments on synthetic data sets and a real-world single-cell transcriptomic data show that RHS outperforms Minimum Spanning Tree (MST), Neighbour Joining, and vanilla HyperSteiner (HS). In near-boundary configurations, RHS can achieve a 32% reduction in total length over HS, demonstrating its effectiveness and robustness in diverse data regimes.

翻译：本文研究双曲空间中Steiner最小树的构建问题。精确SMT计算是NP难的，现有双曲启发式算法（如HyperSteiner）具有确定性，常陷入局部次优构型。我们提出随机化HyperSteiner，这是一种随机Delaunay三角剖分启发式算法，通过在扩展过程中引入随机性，并利用黎曼梯度下降优化对候选树进行细化。在合成数据集和真实单细胞转录组数据上的实验表明，RHS在性能上优于最小生成树、邻接连接法及原始HyperSteiner算法。在近边界构型中，RHS相比HS可实现总长度32%的缩减，证明了其在多样化数据体系中的有效性与鲁棒性。