Breaking the Curse of Dimensionality: On the Stability of Modern Vector Retrieval

2025-12-13

AI Summary

High-dimensional vector retrieval suffers from “distance concentration,” causing unstable nearest-neighbor (NN) results: minor query perturbations trigger drastic changes in the NN set. This paper proposes a **stability-centric unified analytical framework**, the first to systematically address three mainstream retrieval paradigms: (1) multi-vector retrieval—deriving fidelity conditions for stability under Chamfer distance; (2) filtered retrieval—revealing how large penalty mechanisms inherently promote stability; and (3) sparse vector retrieval—establishing a novel sufficient stability criterion under structured sparsity constraints. Through rigorous theoretical analysis, principled modeling, and extensive validation on both synthetic and real-world datasets, our theoretical predictions align closely with empirical observations. The results yield verifiable, actionable stability guarantees for vector index design, embedding training, and retrieval-augmented generation (RAG) systems.

Technology Category

• Computer Vision: Image and Video Retrieval
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Evaluation and Analysis

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Modern vector databases enable efficient retrieval over high-dimensional neural embeddings, powering applications from web search to retrieval-augmented generation. However, classical theory predicts such tasks should suffer from the curse of dimensionality, where distances between points become nearly indistinguishable, thereby crippling efficient nearest-neighbor search. We revisit this paradox through the lens of stability, the property that small perturbations to a query do not radically alter its nearest neighbors. Building on foundational results, we extend stability theory to three key retrieval settings widely used in practice: (i) multi-vector search, where we prove that the popular Chamfer distance metric preserves single-vector stability, while average pooling aggregation may destroy it; (ii) filtered vector search, where we show that sufficiently large penalties for mismatched filters can induce stability even when the underlying search is unstable; and (iii) sparse vector search, where we formalize and prove novel sufficient stability conditions. Across synthetic and real datasets, our experimental results match our theoretical predictions, offering concrete guidance for model and system design to avoid the curse of dimensionality.

Problem

• Analyzes stability of vector retrieval in high-dimensional spaces
• Extends stability theory to multi-vector, filtered, and sparse search
• Provides guidance to avoid the curse of dimensionality

Innovation

• Extends stability theory to multi-vector search with Chamfer distance
• Shows filtered search gains stability via large filter penalties
• Formalizes stability conditions for sparse vector search