Geodesic Semantic Search: Cartographic Navigation of Citation Graphs with Learned Local Riemannian Maps

TL;DR

Geodesic Semantic Search (GSS) introduces a geometry-aware citation graph retrieval system that learns local Riemannian metrics to improve semantic search accuracy and efficiency.

Contribution

The paper proposes a novel method for citation graph retrieval using learned local Riemannian metrics, outperforming Euclidean-based methods in accuracy and computational efficiency.

Findings

Achieves 23% relative improvement in Recall@20 over baselines.

Provides theoretical guarantees for geodesic retrieval advantages.

Reduces computational cost by 4x with minimal loss in retrieval quality.

Abstract

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.