On the Difficulty of Nearest Neighbor Search

TL;DR

This paper introduces Relative Contrast, a new measure to evaluate the inherent difficulty of approximate nearest neighbor search in datasets, analyzing how data properties influence search complexity and explaining the effectiveness of certain heuristic algorithms.

Contribution

It presents the first concrete difficulty measure for approximate NN search, linking data characteristics to search complexity and analyzing existing heuristics within this framework.

Findings

Relative Contrast quantifies dataset difficulty for NN search.

Difficulty measure influences the complexity of Local Sensitive Hashing.

Existing heuristics are explained as special cases of the measure.

Abstract

Fast approximate nearest neighbor (NN) search in large databases is becoming popular. Several powerful learning-based formulations have been proposed recently. However, not much attention has been paid to a more fundamental question: how difficult is (approximate) nearest neighbor search in a given data set? And which data properties affect the difficulty of nearest neighbor search and how? This paper introduces the first concrete measure called Relative Contrast that can be used to evaluate the influence of several crucial data characteristics such as dimensionality, sparsity, and database size simultaneously in arbitrary normed metric spaces. Moreover, we present a theoretical analysis to prove how the difficulty measure (relative contrast) determines/affects the complexity of Local Sensitive Hashing, a popular approximate NN search method. Relative contrast also provides an explanation for a family of heuristic hashing algorithms with good practical performance based on PCA. Finally, we show that most of the previous works in measuring NN search meaningfulness/difficulty can be derived as special asymptotic cases for dense vectors of the proposed measure.