Breaking the Curse of Dimensionality: On the Stability of Modern Vector Retrieval (2512.12458v1)

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.

• The paper establishes a principled framework proving that Chamfer distance ensures stability in multi-vector retrieval, while average pooling may induce instability.
• The paper demonstrates that appropriately tuned filter penalties guarantee nearest neighbor stability in hybrid vector search by overcoming high-dimensional challenges.
• The paper shows that sparse vector search leverages concentration of importance and overlapping supports to achieve robust and scalable high-dimensional retrieval.

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

Introduction

This paper, "Breaking the Curse of Dimensionality: On the Stability of Modern Vector Retrieval" (2512.12458), rigorously analyzes why sub-linear vector retrieval algorithms continue to perform effectively on high-dimensional neural embeddings, seemingly defying classical predictions of the curse of dimensionality. The authors extend foundational theoretical frameworks on vector search stability and empirically demonstrate their applicability to three dominant retrieval paradigms: multi-vector search, filtered vector search, and sparse vector search. Their work prescribes concrete, verifiable criteria for practitioners to ensure stable, efficient retrieval, providing actionable guidance beyond intrinsic dimensionality heuristics.

Theoretical Foundations

The curse of dimensionality implies that, as data dimensionality increases, pointwise distances concentrate, resulting in vanishing contrast necessary for meaningful nearest neighbor (NN) search. The classical formalization by Beyer et al. (1999) and Durrant & Kabán (2009) relates instability to the limiting behavior of the "relative variance" of pointwise distances. If, as dimensionality grows, the ratio of the variance to the squared expectation of the distance metric collapses to zero, retrieval is deemed unstable; NN selection becomes arbitrary and brute-force search is unavoidable.

The authors refocus analysis from geometric peculiarities to "stability": small perturbations to a query should not radically reorder its NN list. Using this formalism, they analyze the stability in three modern retrieval contexts:

• Multi-vector retrieval (e.g., ColBERT-style late-interaction),
• Filtered (hybrid) vector search (vector+metadata constraints),
• Sparse vector search (SPLADE-type, high-dim but with few non-zeros).

Multi-Vector Search Stability

The paper generalizes classical stability results to aggregation functions over sets of vectors. For a collection of query and document vector sets, with a set-aggregation metric $\mathrm{Agg}$ (e.g., Chamfer distance or average pooling), the authors formalize multi-vector stability and introduce the concepts of induced single-vector search and strong stability.

Main Theoretical Results

• Chamfer distance preserves stability: If the induced single-vector problem is "strongly stable" (min-max distance gap bounded away from one), then, under reasonable non-degeneracy and weak intervector covariance assumptions, the Chamfer-aggregated multi-vector search is stable (Theorem 5.9).
• Average pooling does **not guarantee stability**: They demonstrate via construction that average pooling can collapse contrast and induce instability, even when the underlying single-vector problem is stable.

Empirical studies on real (ColBERT) and synthetic datasets confirm these claims: stability ratios and contrast in Chamfer-aggregated searches persist at scale, whereas average pooling rapidly fails.

Filtered Vector Search Stability

This section analyzes the effect of discrete attribute filters (often implemented via additive penalties for predicate mismatch) on NN stability. The penalty model is both expressive and representative for most known hybrid search systems.

Main Theoretical Result

• Sufficiently large penalties induce stability: If the penalty for a filter mismatch is set above a dataset-dependent threshold (scaling with maximal interpoint distance and attribute mismatch probability), filtered search is stable—even if the underlying unconstrained vector search is unstable (Theorem 6.3). This is theoretically robust to attribute and embedding distributions.

Experiments corroborate that below-threshold penalties are insufficient, but once the threshold is crossed, instability is eliminated. This has practical implications for the configuration of hybrid vector retrieval systems.

Sparse Vector Search Stability

Sparse retrieval scenarios involve extremely large embedding dimensions with only a small fraction of nonzero coordinates (e.g., SPLADE, inverted indices). Prior work (Bruch et al., 2024) highlighted empirical advantages of "concentration of importance": the majority of the $\ell_1$ or $\ell_2$ mass is carried by a small head subset of coordinates.

Main Theoretical Results

• Concentration of importance (CoI) and overlap of importance are sufficient for stability: The authors prove that CoI, along with a nontrivial intersection probability between the support of high-concentration query/document heads, guarantees positivity of the limiting relative variance, hence stability (Theorem 7.4).
• Empirically, both CoI and overlap must be present: If high-mass coordinates are concentrated but non-overlapping between queries and database entries, distance collapse (and thus instability) persists. This provides a formal criterion for the design of sparse encodings and support selection in retrieval pipelines.

Validation on real SPLADE embeddings (over 30K dimensions) from several IR benchmarks consistently confirms that both theoretical preconditions are satisfied, resulting in high stability ratios.

Implications and Future Directions

The findings have multiple theoretical and practical repercussions:

• System design: The results furnish constructive constraints for model and index engineering (e.g., Chamfer over mean-pooling for multi-vector, formal penalty tuning for hybrid search, architectural bias toward overlap for sparse models).
• Beyond intrinsic dimensionality: Merely reducing intrinsic dimensionality is insufficient for robust retrieval; the nature of aggregation, filtering, and structure of support critically determine stability.
• Scalability: Since LLM-induced and transformer-derived embeddings increase in size, understanding and maintaining these conditions is essential for scalable, low-latency retrieval.

Prospective work could generalize the mathematical criteria for stability to a broader array of data modalities, alternative aggregations, or learned filter functions. Additionally, there is scope to investigate how representation learning objectives interact with stability directly, possibly designing architectures or losses provably conducive to stable retrieval.

Conclusion

This paper establishes a principled, actionable framework to explain and guarantee the efficiency of modern vector retrieval systems under high-dimensionality. By extending the notion of stability to multi-vector, filtered, and sparse embedding regimes, and rigorously demonstrating when and how the curse of dimensionality can be sidestepped, the work provides both deep theoretical insights and concrete recommendations for practitioners aiming to deploy robust, scalable vector search technology (2512.12458).