Stable Multi-Vector Search

• Multi-vector search stability is defined as the system's ability to maintain consistent performance and recall despite perturbations in vector embeddings and filter conditions.
• Algorithmic strategies such as FCVI, UNIFY, and NaviX integrate geometric transformations and adaptive index structures to mitigate performance degradation under varying selectivity.
• Empirical evaluations demonstrate that methods like FCVI-HSW and Curator achieve minimal latency and recall loss, ensuring robust production-scale retrieval.

Multi-vector search stability refers to the robustness and consistency of performance, accuracy, and resource efficiency in vector search systems that support complex filtering conditions or joint querying over high-dimensional data and structured predicates. Stability in this context encompasses both theoretical resilience—such as guaranteed preservation of recall and bounded degradation under distribution shifts—and empirical metrics such as flat latency curves, bounded recall loss, and memory/resource efficiency as dataset characteristics or filter patterns vary. This property is critical for production-scale retrieval systems, where high-dimensional embeddings are combined with attribute filtering (categorical, range, or complex predicates) under dynamically evolving workloads and distributions.

1. Mathematical Foundations of Stability in Filtered Vector Search

Stability in filtered and multi-vector search is founded on the principle that minor changes to either the vector embeddings or filter predicates should not induce disproportionate changes in retrieval quality or system efficiency. A key formalism is the ε-stability definition: for a filtered search on dataset $D = \{(d_i, A_i^D)\}$ and query set $Q = \{(q_j, A_j^Q)\}$, with base distance $d(q, d)$, filters $f(A^Q, A^D)$, and additive penalty λ, the filtered distance is

$$d_f(q, d) = d(q, d) + \lambda \mathbf{1}_{f(A^Q, A^D) = 0}.$$

ε-stability demands that for any perturbation of the query vector by at most ε in norm, the top-k nearest neighbors under $d_f$ remain unchanged. A sufficient condition is that the minimum gap between the closest accepted and rejected points exceeds twice the largest possible perturbation-induced change, i.e., the filter penalty λ must be chosen to exceed this gap.

The theory further asserts, and demonstrates formally, that filtered search with an appropriate λ avoids the curse of dimensionality that plagues pure vector search. Specifically, as dimensionality increases and plain vector distances collapse (become almost indistinguishable), the injection of a filter penalty re-establishes spread among relevant and irrelevant points. The filtered distance’s relative variance

$$\mathrm{RelVar}(d_f) = \frac{\mathrm{Var}_q(d_f(q, D))}{(\mathbb{E}_q[d_f(q, D)])^2}$$

stays bounded away from zero if

$\lambda > \frac{4A}{1-p_{\max}},$

where A is the maximum base distance and $p_{\max}$ is the maximum probability of a query–document filter match (Lakshman et al., 13 Dec 2025). This guarantees stability—namely, sublinear search remains effective and robust even as data dimensionality grows.

2. Algorithmic Strategies for Stable Multi-Vector Search

A broad range of algorithmic frameworks deliver stability in multi-vector and filtered vector search:

• Geometric Transformation (FCVI): Encodes filters directly into embedding space by applying a transformation $\psi(v, f, \alpha)$ defined by

$$\psi(v, f, \alpha) = [v^{(1)} - \alpha f, ..., v^{(d/m)} - \alpha f]$$

for partitioned vectors, with α controlling filter influence. This technique ensures intra-filter distances are preserved while inter-filter distances increase as α grows, yielding strong stability—and recall guarantees—for any fixed vector index (Heidari et al., 19 Jun 2025).

• Index Set Construction (SIEVE): Builds a small, workload-aware collection of predicate-specific indexes, each optimized for a particular selectivity range. Query-time logic selects the optimal index based on query selectivity, guided by a three-dimensional model of memory, latency, and recall. This diversity and explicit analytical modeling ensure search time and accuracy remain stable as predicate selectivity changes (Li et al., 16 Jul 2025).
• Dual/Hierarchical Index Partitioning (Curator, UNIFY): Combines graph or hierarchical tree indexes (e.g., hierarchical k-means trees or segmented graphs) for general queries with lightweight, adaptive structures for low-selectivity filters. The transition between structures is automatic given current selectivity, yielding near-constant latency across regimes where connectivity or graph traversal would otherwise degrade (Jin et al., 3 Jan 2026, Liang et al., 2024).
• Predicate-agnostic Adaptive Traversal (NaviX): Prefilters candidate sets with DBMS-level mask computation and dynamically adapts HNSW graph traversal according to local filter selectivity, optimizing for stability across a spectrum of predicate correlations and selectivity regimes (Sehgal et al., 29 Jun 2025).

3. Theoretical Guarantees and Bounds

Many contemporary frameworks provide provable guarantees for stability:

• Filter-Centric Vector Indexing (FCVI) Theorems:

1. For α ≥ 1, filter mismatch terms dominate the separation, quadratically increasing distance between different filter classes while preserving intra-class geometry.
2. The retrieval efficiency theorem shows that, for scoring rule score = λ sim_v + (1–λ) sim_f, one needs only O(k·(1/λ)·(1/α²)) candidates to guarantee top-k recall.
3. Uniqueness theorem states that any linear, symmetric transformation achieving these properties must be of the FCVI form up to constants.
4. Cluster separation is explicitly characterized by the minimum inter-filter gap and maximum intra-filter variance, yielding closed-form expressions for α* that guarantee no overlap across filter-separated clusters (Heidari et al., 19 Jun 2025).

• Penalty-based Filter Stability: For arbitrary high dimensions, if the filter penalty λ exceeds the collapse threshold as above, retrieval “RecVar” remains strictly positive, and nearest neighbor identities are unaffected by small query perturbations (Lakshman et al., 13 Dec 2025).
• Pareto Frontier in SIEVE: The analytical model ensures that, at any recall target and memory budget, SIEVE deploys the index configuration yielding the best achievable latency, mathematically bounding worst-case deviation from optimal as selectivity shifts (Li et al., 16 Jul 2025).

These theoretical underpinnings provide strict worst-case and average-case stability results, quantifying both recall preservation and resource usage sensitivity.

4. Empirical Evaluation and Comparative Outcomes

Across diverse vector search systems, empirical results consistently validate the theory. Representative findings include:

Method	Latency Change (Shift)	Recall Drop (Shift)	Stability Classification
FCVI-HSW	<25%	<2.5%	High
UNIFY	~42–56%	~3.8–5.1%	Medium
Pre-filter	>80%	up to –15%	Very Low
Post-filter	>60%	up to –7.6%	Low

• FCVI-HSW: Under distribution shifts (e.g., in filter selectivity, vector clusters, or query patterns), latency and recall remain almost flat (<25% latency increase, <2.5% recall loss), outpacing traditional pre/post-filtering and hybrid techniques (Heidari et al., 19 Jun 2025).
• Curator: For selectivities in s ∈ [0.001, 0.2], latency varies by <2× (versus >100× for graph-based methods) thanks to hierarchical clustering, with negligible memory and build-time overhead (<5–10%) (Jin et al., 3 Jan 2026).
• UNIFY-HSIG: Supports both pre-, post-, and hybrid filtering in a unified index, yielding a QPS curve that varies by less than 10% across the entire selectivity spectrum, while recall remains at the target level (Liang et al., 2024).
• SIEVE: Maintains throughput speedups between 1.1× to 3.6× over filtered baselines across all tested selectivities, with controlled recall and modest (30–80%) memory overhead, reflecting fine-grained theoretical control over T(S,R) (Li et al., 16 Jul 2025).
• NaviX: Demonstrates nearly flat latency from σ = 90% to 1% (e.g., 5 ms at high σ to 7 ms at low σ for Wiki-15M), outperforming both pre- and post-filtering by up to 100× at low selectivity (Sehgal et al., 29 Jun 2025).

5. Mechanisms Preventing Degradation Under Distribution Shifts

Instability in earlier or naïve approaches is primarily due to either loss of connectivity in filtered graphs (pre-filtering), high false positives in post-filtering, or suboptimal segment-specific data statistics as query or data distributions drift. Stable methods counter these pitfalls:

• Unified Geometric Embedding (FCVI): Encodes filter and vector relations such that their structure persists under both filter and vector distributional changes, requiring no re-partitioning.
• Inclusive/Hierarchical Index Structures (UNIFY, Curator): Dynamically adjust to varying predicate selectivity without structural rebuilding or non-monotonic performance loss.
• Predicate-Agnostic Traversal (NaviX): Local selectivity estimation steers the algorithm between 1-hop, 2-hop, or sorted traversal, sidestepping the performance collapse of any single fixed heuristic.
• Index Collection and Analytical Control (SIEVE): Explicitly tracks the cost/recall trade-off over the predicate selectivity axis, ensuring that typical queries always use an optimal dedicated index.

These mechanisms are empirically validated to preserve high recall (>90–95%) and nearly stable latency/QPS profiles even as selectivities, filter cardinalities, and data distributions vary widely.

6. Practical Integration and Limitations

Stable multi-vector search frameworks introduce new integration patterns:

• FCVI: Transformed vectors are constructed offline and indexed in standard ANN libraries (HNSW/FAISS/ANNOY) with no index library changes. Query-side transformation and re-scoring are lightweight, with parameter tuning for λ, α dictated by theory, and only rare need for one-index-per-field trade-offs (Heidari et al., 19 Jun 2025).
• UNIFY, Curator: Require only a single unified or shared tree/graph structure; per-label or per-segment indexes are embedded, not duplicated, limiting resource overhead. Complex predicates are supported ad hoc via temporary index construction (Liang et al., 2024, Jin et al., 3 Jan 2026).
• SIEVE: Pre-computes a small collection of indexes, with at-query overhead limited to a histogram/bin lookup and index selector (Li et al., 16 Jul 2025).
• NaviX: Tight integration with underlying DBMS buffer and predicate evaluation enables zero-copy distance computation and full predicate-agnosticity (Sehgal et al., 29 Jun 2025).

Identified limitations include the need for separate indexes per independent filter-combination in FCVI (for pure isolation), as well as the reliance on accurate selectivity estimation or workload sampling for optimal packing in SIEVE. Most reported empirical evaluations concern medium-scale, metadata-rich corpora; further validation on billion-scale data is noted as an area for solidifying production readiness.

7. Outlook and Broader Implications

Multi-vector search stability is now characterized by rigorous theoretical bounds that link geometric or penalty-based mechanisms with robust empirical performance. State-of-the-art systems emphasize unified, adaptable infrastructure—either via embedding, dual-indexing, or analytical index collection—that decouples system stability from unpredictable workload shifts and data distribution changes. This suggests a paradigm where stable performance is not a coincidental artifact of favorable distributions but an engineered, provable property. The field currently anticipates future work in scaling these techniques to web-scale datasets and complex, composite predicate workloads with the same level of provable stability and empirical robustness (Heidari et al., 19 Jun 2025, Lakshman et al., 13 Dec 2025, Perini et al., 2024, Liang et al., 2024, Li et al., 16 Jul 2025, Jin et al., 3 Jan 2026, Sehgal et al., 29 Jun 2025).