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Geodesic Semantic Search: Learning Local Riemannian Metrics for Citation Graph Retrieval

Brandon Yee, Lucas Wang, Kundana Kommini, Krishna Sharma

TL;DR

A retrieval system that learns node-specific Riemannian metrics on citation graphs to enable geometry-aware semantic search, and theoretical analysis of when geodesic distances outperform direct similarity, characterize the approximation quality of low-rank metrics, and validate predictions empirically are provided.

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 papers, \gss{} achieves 23\% relative improvement in Recall@20 over SPECTER+FAISS baselines while providing interpretable citation paths. Our hierarchical coarse-to-fine search with k-means pooling reduces computational cost by 4$\times$ compared to flat geodesic search while maintaining 97\% retrieval quality. We provide theoretical analysis of when geodesic distances outperform direct similarity, characterize the approximation quality of low-rank metrics, and validate predictions empirically. Code and trained models are available at https://github.com/YCRG-Labs/geodesic-search.

Geodesic Semantic Search: Learning Local Riemannian Metrics for Citation Graph Retrieval

TL;DR

A retrieval system that learns node-specific Riemannian metrics on citation graphs to enable geometry-aware semantic search, and theoretical analysis of when geodesic distances outperform direct similarity, characterize the approximation quality of low-rank metrics, and validate predictions empirically are provided.

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 at each node, inducing a local positive semi-definite metric . 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 papers, \gss{} achieves 23\% relative improvement in Recall@20 over SPECTER+FAISS baselines while providing interpretable citation paths. Our hierarchical coarse-to-fine search with k-means pooling reduces computational cost by 4 compared to flat geodesic search while maintaining 97\% retrieval quality. We provide theoretical analysis of when geodesic distances outperform direct similarity, characterize the approximation quality of low-rank metrics, and validate predictions empirically. Code and trained models are available at https://github.com/YCRG-Labs/geodesic-search.
Paper Structure (68 sections, 5 theorems, 27 equations, 2 figures, 12 tables, 1 algorithm)

This paper contains 68 sections, 5 theorems, 27 equations, 2 figures, 12 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related Work
  4. Geometric Deep Learning.
  5. Hyperbolic Embeddings.
  6. Graph Neural Networks for Retrieval.
  7. Dense Retrieval.
  8. Metric Learning.
  9. Problem Formulation
  10. Low-Rank Metric Parameterization
  11. Method
  12. MetricGAT Architecture
  13. Graph Attention Layers.
  14. Embedding Head.
  15. Metric Head.
  16. ...and 53 more sections

Key Result

Proposition 1

For any $\mathbf{L}_i \in \mathbb{R}^{d \times r}$ with $r \leq d$ and $\epsilon > 0$, the matrix is symmetric positive definite with minimum eigenvalue at least $\epsilon$.

Figures (2)

  • Figure 1: MetricGAT architecture. Input SPECTER features are processed through graph attention layers, then split into embedding and metric heads.
  • Figure 2: Visualization of learned local metrics. (a) Metric ellipsoids vary by field. (b) Metric variance correlates with local graph density.

Theorems & Definitions (12)

  • Definition 1: Local Riemannian Metric
  • Definition 2: Graph Geodesic Distance
  • Proposition 1: Low-Rank Metric Guarantee
  • proof
  • Proposition 2: Hierarchical Approximation
  • Theorem 3: Geodesic Advantage Condition
  • proof : Proof sketch
  • Theorem 4: Low-Rank Metric Approximation
  • Proposition 5: Smoothness-Coherence Connection
  • proof
  • ...and 2 more