δ-EMG: A Monotonic Graph Index for Approximate Nearest Neighbor Search

TL;DR

This work addresses the lack of universal error guarantees in high-dimensional ANN by introducing the $\delta$-Error-Bounded Monotonic Graph ($\delta$-EMG) that guarantees a $1/\delta$-approximation for top-$k$ ANN queries via monotonic navigation. It develops both an exact and a scalable approximate construction, and further introduces a quantized variant, $\delta$-EMQG$, with an adaptive edge scheme and RaBitQ-based distance estimation. The proposed methods demonstrate strong empirical performance, achieving up to $19{,}000$ QPS on SIFT1M at 99% recall and showing robust scalability and provable error-bounded guarantees, outperforming leading baselines across multiple datasets. These results offer a practical, provably reliable alternative for real-world large-scale ANN systems, combining theoretical guarantees with high throughput. Overall, the paper advances the design of graph-based ANN indexes by coupling monotonic navigability with adaptive construction and quantization while maintaining strict error bounds.

Abstract

Approximate nearest neighbor (ANN) search in high-dimensional spaces is a foundational component of many modern retrieval and recommendation systems. Currently, almost all algorithms follow an $ε$-Recall-Bounded principle when comparing performance: they require the ANN search results to achieve a recall of more than $1-ε$ and then compare query-per-second (QPS) performance. However, this approach only accounts for the recall of true positive results and does not provide guarantees on the deviation of incorrect results. To address this limitation, we focus on an Error-Bounded ANN method, which ensures that the returned results are a $(1/δ)$-approximation of the true values. Our approach adopts a graph-based framework. To enable Error-Bounded ANN search, we propose a $δ$-EMG (Error-bounded Monotonic Graph), which, for the first time, provides a provable approximation for arbitrary queries. By enforcing a $δ$-monotonic geometric constraint during graph construction, $δ$-EMG ensures that any greedy search converges to a $(1/δ)$-approximate neighbor without backtracking. Building on this foundation, we design an error-bounded top-$k$ ANN search algorithm that adaptively controls approximation accuracy during query time. To make the framework practical at scale, we introduce $δ$-EMQG (Error-bounded Monotonic Quantized Graph), a localized and degree-balanced variant with near-linear construction complexity. We further integrate vector quantization to accelerate distance computation while preserving theoretical guarantees. Extensive experiments on the ANN-Benchmarks dataset demonstrate the effectiveness of our approach. Under a recall requirement of 0.99, our algorithm achieves 19,000 QPS on the SIFT1M dataset, outperforming other methods by more than 40\%.

Figures (10)

• Figure 1: A comparison of Monotonic Graph and $\delta\text{-}{ {\mathsf{EMG}}}$. $v_1, \dots, v_7$ are nodes in the graph and $q$ is the query. The red edges indicate additional links introduced in $\delta\text{-}{ {\mathsf{EMG}}}$ compared to the Monotonic Graph.
• Figure 2: The construction rule for $\delta\text{-}{ {\mathsf{EMG}}}$. The gray shaded area represents the Occlusion Region for $(u,v)$. For any query $q$ inside blue shaded area, $u$ is guaranteed to have a neighbor closer to $q$.
• Figure 3: QPS vs Recall
• Figure 4: Construction Cost
• Figure 5: Effect of $\delta$

Theorems & Definitions (16)

• Definition 2: top-$k$ $\epsilon$-Recall-Bounded ANN Search
• Definition 3: top-$k$ $\delta$-Error-Bounded ANN Search
• Definition 4: Monotonic Path
• Definition 5: Monotonic Graph
• Definition 6: Monotonic Top-1 Search
• Theorem 1
• Example 1
• Definition 7: $\delta$-Neighborhood
• Definition 8: $\delta$-Error-Bounded Monotonic Graph
• Example 2