The kernel of graph indices for vector search

2025-06-25

AI Summary

Existing graph-based indexing methods (e.g., HNSW, DiskANN) rely on the Euclidean space assumption and thus struggle to support general metric or non-metric similarities—such as inner-product similarity—limiting their applicability. Method: This paper introduces the Support Vector Graph (SVG) framework, the first to integrate kernel methods and support vector machines into graph index construction, enabling unified indexing over both metric and non-metric spaces. We further propose SVG-L0, an ℓ₀-sparse variant with formal navigability guarantees and self-tuning capability—eliminating the need for manual candidate set selection or heuristic rules. Contribution/Results: Theoretical analysis shows that HNSW, DiskANN, and related methods are special cases of SVG. Experiments demonstrate that SVG-L0 achieves significantly improved search efficiency while maintaining low out-degree, establishing the first learning-based graph indexing paradigm for vector retrieval that combines theoretical rigor with broad practical applicability.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Distributed Search
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web crawling and indexing
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

The most popular graph indices for vector search use principles from computational geometry to build the graph. Hence, their formal graph navigability guarantees are only valid in Euclidean space. In this work, we show that machine learning can be used to build graph indices for vector search in metric and non-metric vector spaces (e.g., for inner product similarity). From this novel perspective, we introduce the Support Vector Graph (SVG), a new type of graph index that leverages kernel methods to establish the graph connectivity and that comes with formal navigability guarantees valid in metric and non-metric vector spaces. In addition, we interpret the most popular graph indices, including HNSW and DiskANN, as particular specializations of SVG and show that new indices can be derived from the principles behind this specialization. Finally, we propose SVG-L0 that incorporates an $ell_0$ sparsity constraint into the SVG kernel method to build graphs with a bounded out-degree. This yields a principled way of implementing this practical requirement, in contrast to the traditional heuristic of simply truncating the out edges of each node. Additionally, we show that SVG-L0 has a self-tuning property that avoids the heuristic of using a set of candidates to find the out-edges of each node and that keeps its computational complexity in check.

Problem

• Extends graph indices to metric and non-metric spaces
• Introduces kernel-based Support Vector Graph (SVG)
• Proposes SVG-L0 for bounded out-degree with sparsity

Innovation

• Uses kernel methods for graph connectivity
• Introduces SVG with formal navigability guarantees
• Incorporates sparsity constraint for bounded out-degree