The Geometry of Hierarchical Navigation: Accuracy and Query Cost for Point Process Input

2026-10-08

AI Summary

This study addresses the lack of geometric theoretical foundations for hierarchical navigation in high-dimensional vector spaces within large-scale retrieval systems, which limits the accuracy and efficiency of approximate nearest neighbor search. To this end, this work investigates greedy search on torus proximity graphs, employing stochastic geometry and point process theory—including Poisson, determinantal, and Cox processes—to conduct rigorous mathematical analysis. The primary contribution is the first identification of deterministic geometric covering conditions that guarantee (1+ε)-approximation accuracy, along with proofs that these conditions hold with high probability under random processes. Furthermore, this research establishes a query cost model demonstrating that, in fixed dimensions, the expected number of hops grows logarithmically with data size. These findings provide a solid theoretical foundation for efficient hierarchical navigation in large-scale retrieval systems.

Abstract

Large-scale information retrieval systems, including retrieval-augmented generation (RAG) and recommendation engines, widely use multi-layered hierarchical data structures for ultra-fast approximate nearest-neighbor search in high-dimensional vector spaces. However, the geometric conditions that ensure accurate and efficient greedy navigation remain poorly understood. In this work, we study the efficiency of greedy navigation on a hierarchy of proximity graphs constructed from \(n\) data points on the \(d\)-dimensional torus~$\mathbb{T}^d$. We identify a deterministic coverage condition under which, given any query $q\in \mathbb{T}^d$, greedy search returns a point within $(1+\varepsilon)$-factor of the distance to the closest point. This coverage property holds with high probability when the data is distributed as a homogeneous Poisson process, a Hermitian determinantal process, or a bounded-density Cox process, as long as $d = o(\log n/\log \log n)$. Under the same assumptions, the expected number of greedy hops for a fixed query is \(O\!\left(\exp\!\left(\tfrac12 d\log d+O(d)\right)\log n\right)\), yielding logarithmic expected hop count in fixed dimension.

Problem

• hierarchical navigation
• approximate nearest-neighbor search
• greedy search
• proximity graphs
• high-dimensional vector spaces

Innovation

• hierarchical navigation
• greedy search
• proximity graphs
• point process
• approximate nearest-neighbor