Riemannian and Symplectic Geometry for Hierarchical Text-Driven Place Recognition

Text-to-point-cloud localization enables robots to understand spatial positions through natural language descriptions, which is crucial for human-robot collaboration in applications such as autonomous driving and last-mile delivery. However, existing methods employ pooled global descriptors for similarity retrieval, which suffer from severe information loss and fail to capture discriminative scene structures. To address these issues, we propose SympLoc, a novel coarse-to-fine localization framework with multi-level alignment in the coarse stage. Different from previous methods that rely solely on global descriptors, our coarse stage consists of three complementary alignment levels: 1) Instance-level alignment establishes direct correspondence between individual object instances in point clouds and textual hints through Riemannian self-attention in hyperbolic space; 2) Relation-level alignment explicitly models pairwise spatial relationships between objects using the Information-Symplectic Relation Encoder (ISRE), which reformulates relation features through Fisher-Rao metric and Hamiltonian dynamics for uncertainty-aware geometrically consistent propagation; 3) Global-level alignment synthesizes discriminative global descriptors via the Spectral Manifold Transform (SMT) that extracts structural invariants through graph spectral analysis. This hierarchical alignment strategy progressively captures fine-grained to coarse-grained scene semantics, enabling robust cross-modal retrieval. Extensive experiments on the KITTI360Pose dataset demonstrate that SympLoc achieves a 19% improvement in Top-1 recall@10m compared to existing state-of-the-art approaches.