Next Point-of-Interest (POI) recommendation is a critical task in modern Location-Based Social Networks (LBSNs), aiming to model the complex decision-making process of human mobility to provide personalized recommendations for a user's next check-in location. Existing hyperbolic POI recommendation models, predominantly based on rotations and graph representations, have been extensively investigated. Although hyperbolic geometry has proven superior in representing hierarchical data with low distortion, current hyperbolic sequence models typically rely on performing recurrence via expensive Möbius operations directly on the manifold. This incurs prohibitive computational costs and numerical instability, rendering them ill-suited for trajectory modeling. To resolve this conflict between geometric representational power and sequential efficiency, we propose GTR-Mamba, a novel framework featuring Geometry-to-Tangent Routing. GTR-Mamba strategically routes complex state transitions to the computationally efficient Euclidean tangent space. Crucially, instead of a static approximation, we introduce a Parallel Transport (PT) mechanism that dynamically aligns tangent spaces along the trajectory. This ensures geometric consistency across recursive updates, effectively bridging the gap between the curved manifold and linear tangent operations. This process is orchestrated by an exogenous spatio-temporal channel, which explicitly modulates the SSM discretization parameters. Extensive experiments on three real-world datasets demonstrate that GTR-Mamba consistently outperforms state-of-the-art baselines in next POI recommendation.

翻译：下一兴趣点推荐是现代基于位置的社交网络中的一项关键任务，其旨在对人类移动的复杂决策过程进行建模，从而为用户的下一个签到位置提供个性化推荐。现有的双曲兴趣点推荐模型主要基于旋转和图表示，已得到广泛研究。尽管双曲几何已被证明在低失真表示层次化数据方面具有优越性，但当前的双曲序列模型通常依赖于直接在流形上通过昂贵的Möbius运算执行递推。这会导致极高的计算成本和数值不稳定性，使其不适合用于轨迹建模。为解决几何表示能力与序列效率之间的这一矛盾，我们提出了GTR-Mamba，这是一个具有几何到切空间路由功能的新型框架。GTR-Mamba策略性地将复杂的状态转移路由到计算高效的欧几里得切空间。关键的是，我们引入了一种平行传输机制，而非静态近似，该机制能沿轨迹动态对齐切空间。这确保了递归更新过程中的几何一致性，有效弥合了弯曲流形与线性切空间操作之间的差距。此过程由一个外生的时空通道协调，该通道显式地调制SSM离散化参数。在三个真实世界数据集上的大量实验表明，GTR-Mamba在下一兴趣点推荐任务中始终优于最先进的基线方法。