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

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 $(1+\epsilon )$-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 $d=o(\log n\mathrm{/}\log \log n)$. Under the same assumptions, the expected number of greedy hops for a fixed query is $O\text{\hspace{0.17em}⁣}(\exp \text{\hspace{0.17em}⁣}(\frac{1}{2}d\log d+O(d))\log n)$, yielding logarithmic expected hop count in fixed dimension.