We present Geodesic Semantic Search (GSS), a retrieval system that learns node-specific Riemannian metrics on citation graphs to enable geometry-aware semantic search. Unlike standard embedding-based retrieval that relies on fixed Euclidean distances, \gss{} learns a low-rank metric tensor $\mL_i \in \R^{d \times r}$ at each node, inducing a local positive semi-definite metric $\mG_i = \mL_i \mL_i^\top + \eps \mI$. This parameterization guarantees valid metrics while keeping the model tractable. Retrieval proceeds via multi-source Dijkstra on the learned geodesic distances, followed by Maximal Marginal Relevance reranking and path coherence filtering. On citation prediction benchmarks with 169K arXiv papers, GSS achieves 23\% relative improvement in Recall@20 over SPECTER+FAISS baselines. We provide a Bridge Recovery Guarantee characterizing when geodesic retrieval qualitatively outperforms direct similarity, a margin separation result connecting training loss to retrieval quality, and characterize the expressiveness of low-rank metric parameterization. Our hierarchical coarse-to-fine search with k-means pooling reduces computational cost by $4\times$ while maintaining 97\% retrieval quality.

翻译：本文提出测地语义检索系统GSS，通过为引文图各节点学习专用黎曼度量实现几何感知语义检索。不同于依赖固定欧氏距离的标准嵌入检索，GSS在每个节点处学习低秩度量张量$\mL_i \in \R^{d \times r}$，生成局部半正定度量$\mG_i = \mL_i \mL_i^\top + \eps \mI$。该参数化方案在保证模型可计算性的同时确保度量有效性。检索过程通过多源Dijkstra算法计算学习得到的测地距离，随后执行最大边界相关重排序与路径一致性过滤。在包含16.9万篇arXiv论文的引文预测基准测试中，GSS在Recall@20指标上较SPECTER+FAISS基线获得23%相对提升。本文提出桥接恢复保证定理，阐明测地检索在何种条件下定性优于直接相似度检索；建立边际分离结果揭示训练损失与检索质量间的关联；并论证低秩度量参数化的表达能力。通过k-means池化实现的分层粗到细搜索策略，在保持97%检索质量的同时将计算成本降低4倍。