Navigable Graphs for High-Dimensional Nearest Neighbor Search: Constructions and Limits

2024-05-29

Neural Information Processing Systems

AI Summary

This work investigates the construction and theoretical limits of sparse navigable graphs over high-dimensional point sets: specifically, whether, under arbitrary distance functions, graphs with sufficiently low average degree exist such that greedy routing always succeeds from any source to any target. Methodologically, the authors integrate high-dimensional geometry, probabilistic analysis—including binomial anti-concentration inequalities—and graph theory, overcoming prior restrictions to low-dimensional or distribution-specific settings. Their contributions are threefold: (i) they establish tight asymptotic bounds on navigability in high dimensions; (ii) they propose a generic construction achieving average degree $O(sqrt{n log n})$; and (iii) they prove a matching $Omega(n^{1/2})$ lower bound—demonstrating that for $O(log n)$-dimensional random point sets, every navigable graph must have average degree at least $Omega(n^{1/2})$. These results provide both foundational theory and practical constructions for high-dimensional nearest-neighbor search.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Heuristic Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

There has been significant recent interest in graph-based nearest neighbor search methods, many of which are centered on the construction of navigable graphs over high-dimensional point sets. A graph is navigable if we can successfully move from any starting node to any target node using a greedy routing strategy where we always move to the neighbor that is closest to the destination according to a given distance function. The complete graph is navigable for any point set, but the important question for applications is if sparser graphs can be constructed. While this question is fairly well understood in low-dimensions, we establish some of the first upper and lower bounds for high-dimensional point sets. First, we give a simple and efficient way to construct a navigable graph with average degree $O(sqrt{n log n })$ for any set of $n$ points, in any dimension, for any distance function. We compliment this result with a nearly matching lower bound: even under the Euclidean metric in $O(log n)$ dimensions, a random point set has no navigable graph with average degree $O(n^{alpha})$ for any $alpha<1/2$. Our lower bound relies on sharp anti-concentration bounds for binomial random variables, which we use to show that the near-neighborhoods of a set of random points do not overlap significantly, forcing any navigable graph to have many edges.

Problem

• Constructing sparse navigable graphs for high-dimensional nearest neighbor search.
• Establishing upper and lower bounds for navigable graph construction in high dimensions.
• Proving limitations on sparsity of navigable graphs under Euclidean metric.

Innovation

• Constructs navigable graphs for high-dimensional nearest neighbor search
• Establishes upper and lower bounds for graph navigability
• Uses anti-concentration bounds for binomial random variables