A Theoretical Analysis Of Nearest Neighbor Search On Approximate Near Neighbor Graph

TL;DR

This paper provides the first theoretical guarantees for greedy nearest neighbor search on approximate near neighbor graphs, addressing a gap between practice and theory in graph-based NN-Search algorithms.

Contribution

It introduces a theoretical framework for greedy search on approximate near neighbor graphs, supported by novel computational geometry tools, for low-dimensional dense vectors.

Findings

Provides guarantees for approximate NN-Search on ANN-Graphs

Quantifies trade-offs in graph construction and search accuracy

Bridges the gap between empirical success and theoretical understanding

Abstract

Graph-based algorithms have demonstrated state-of-the-art performance in the nearest neighbor search (NN-Search) problem. These empirical successes urge the need for theoretical results that guarantee the search quality and efficiency of these algorithms. However, there exists a practice-to-theory gap in the graph-based NN-Search algorithms. Current theoretical literature focuses on greedy search on exact near neighbor graph while practitioners use approximate near neighbor graph (ANN-Graph) to reduce the preprocessing time. This work bridges this gap by presenting the theoretical guarantees of solving NN-Search via greedy search on ANN-Graph for low dimensional and dense vectors. To build this bridge, we leverage several novel tools from computational geometry. Our results provide quantification of the trade-offs associated with the approximation while building a near neighbor graph. We hope our results will open the door for more provable efficient graph-based NN-Search algorithms.