A Tour of Locality Sensitive Filtering on the Sphere

2026-04-27

AI Summary

This work addresses the problem of angular approximate nearest neighbor (Angular ANN) search on high-dimensional spheres by proposing a unified framework that systematically integrates locality-sensitive hashing (LSH) and locality-sensitive filtering (LSF). By constructing a novel LSF-based data structure, the paper reformulates LSF theory in an expository “guided tour” manner, revealing its deep connections with LSH and strengthening key lemmas to establish the optimality of the proposed structure in terms of both query and space complexity. The study not only delivers a concise and rigorous theoretical analysis but also provides a clear, coherent entry point and survey perspective for future research on Angular ANN.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

The Approximate Near Neighbor (ANN) problem is a cornerstone in high-dimensional data analysis, with applications ranging from information retrieval to data mining. Among the most successful paradigms for solving ANN in high-dimensional metric spaces is Locality Sensitive Hashing (LSH), alongside the more recent Locality Sensitive Filtering (LSF). Since the seminal work of Indyk and Motwani, literature has expanded into a complex landscape of variants, often presented under different perspectives and adopting different notation. In this work, we address the technical challenge of navigating this landscape, by providing a self-contained, unified view of the essential algorithmic ingredients governing LSH-based and LSF-based solutions for angular distance -- a case of particular relevance in modern applications. In doing so, we touch on deep connections between LSF and LSH strategies. Our contribution is twofold. First, we design and analyze an LSF-based data structure for the Angular ANN problem, serving as a "guided tour" through fundamental techniques and results in the area. Second, we provide a streamlined analysis that, piecing together technical ingredients and results appeared throughout previous literature, proves optimality of the proposed data structure. In doing so, we revisit and strengthen a key technical lemma central to this body of work. The result is a critical and cohesive review that identifies core mechanisms of proximity search, offering both a streamlined entry point for researchers and a refined perspective on the state of the art.

Problem

• Approximate Near Neighbor
• Locality Sensitive Filtering
• angular distance
• high-dimensional search
• proximity search

Innovation

• Locality Sensitive Filtering
• Approximate Near Neighbor
• Angular Distance
• Optimality Analysis
• High-dimensional Search