Filtered Vector Search Stability

• The paper establishes that stability in filtered vector search is defined by bounded variance of penalized distances, ensuring predictable performance under variable selectivity.
• Methodologies such as adaptive search heuristics, unified multi-strategy indexes, and geometric transformations demonstrate significant latency reductions and throughput gains across diverse query regimes.
• Empirical results show that approaches like Curator and SIEVE achieve orders-of-magnitude improvements in performance consistency, effectively mitigating issues like graph fragmentation and the curse of dimensionality.

Filtered vector search stability concerns the robustness and predictability of approximate nearest neighbor search methods when queries are restricted to subsets of data defined by arbitrary predicates or attribute filters. As modern applications frequently combine vector similarity with attribute-based selection, stability analysis focuses on ensuring that search latency, recall, and resource usage remain well-controlled across the entire spectrum of filter selectivities, query correlations, and distributional shifts.

1. Formalization of Filtered Vector Search and Stability

Filtered vector search generalizes standard k-nearest-neighbor (kNN) search by only considering a subset $S$ of the dataset %%%%1%%%%, where $S$ is determined by some selection predicate or filter. The filtered $k$-NN problem is:

$$\text{Given}\ (V, QS, v_Q, k),\ \text{find}\ \text{kNN}(v_Q, S) = \operatorname{argmin}_u^{k} \mathrm{dist}(v_Q, u),\ u\in S\subseteq V,$$

with $\mathrm{dist}$ denoting a distance metric such as Euclidean or cosine distance (Sehgal et al., 29 Jun 2025).

Stability is defined as the system’s ability to maintain predictable performance (latency, recall, throughput) as selectivity $\sigma = |S|/|V|$ and filter/query correlations vary. In high dimensions, classical instability arises as distances concentrate, but filter integration can recover stability if managed carefully (Lakshman et al., 13 Dec 2025).

Formally, stability in filtered search is sometimes expressed by the boundedness of the relative variance of penalized distances:

$$\liminf_{m\to\infty} \frac{\mathrm{Var}[\delta'(q,d)]}{\mathbb{E}[\delta'(q,d)]^2} > 0,$$

where $\delta'(q,d)$ incorporates a filter-based penalty, ensuring distance spread is preserved (Lakshman et al., 13 Dec 2025).

2. Structural Causes of Instability

Standard graph-based indices (e.g., HNSW) exhibit instability when filters select only a small subset of the dataset. In such cases, the induced subgraph on qualifying nodes may fragment, causing search connectivity breakdown and exploding latency (Jin et al., 3 Jan 2026, Li et al., 16 Jul 2025). Pre-filtering approaches suffer from linear complexity in $|S|$ at low selectivity, while post-filtering methods waste significant computation on non-qualifying candidates in small-$\sigma$ regimes (Liang et al., 2024, Heidari et al., 19 Jun 2025). The curse of dimensionality further exacerbates brittleness, as distances between points become nearly indistinguishable in unfiltered spaces (Lakshman et al., 13 Dec 2025).

3. Stabilization Methodologies

A wide range of strategies have been proposed to achieve stability across all selectivities and workload types:

Adaptive Search Heuristics

NaviX introduces adaptive-local search, leveraging per-node local selectivity $\sigma_\ell(c, S)$ to pick among fixed heuristics (onehop-s, directed, blind) at each step (Sehgal et al., 29 Jun 2025). The exploration strategy dynamically changes based on the density of $S$ around current candidates. This approach provably traces the lower envelope of cost curves across all selectivity and correlation regimes.

Unified Multi-Strategy Indexes

UNIFY constructs a hierarchical segmented inclusive graph (HSIG) that supports pre-filtering, post-filtering, and hybrid-filtered ANNS. An automatic range-aware selector picks among these strategies based on the query’s filter range, keeping latency nearly flat from small (1%) to large (100%) selectivities (Liang et al., 2024).

Partition-Based and Collection Methods

Curator augments graph indexes with a shared clustering tree, embedding specialized per-label sub-indexes and using Bloom filters for rapid label membership tests. This dual-structure ensures logarithmic query cost and prevents performance collapse at low selectivities, achieving up to 20.9× speedup at $s=10^{-3}$ (Jin et al., 3 Jan 2026). SIEVE organizes a workload-aware collection of indexes, each built for a specific predicate template. At query time, the optimal index is selected by an analytical cost model that predicts latency as a function of selectivity and recall, yielding uniformly stable performance (Li et al., 16 Jul 2025).

Geometric Transformation

FCVI encodes filter conditions as a geometric transformation $\psi(v, f, \alpha)$ in the vector embedding space, natively integrating filter separation into the search. Increased $\alpha$ amplifies the effect of the filter, and the method is mathematically shown to preserve recall and stability even as filter patterns or vector distributions shift (Heidari et al., 19 Jun 2025).

Penalized Distance Formulation

Properly tuned additive penalties for filter mismatches (e.g., $$\delta'(q, d) = \delta(q,d) + \alpha \cdot [1 - f(A_e, A_d)]$$) guarantee retention of relative variance and thereby stability, provided $\alpha > 2\Delta/(1-p_{\mathrm{max}})$, with $\Delta$ the maximum vector distance and $p_{\mathrm{max}}$ the filter miss probability (Lakshman et al., 13 Dec 2025).

4. Stability Analysis and Theoretical Guarantees

Theoretical stability guarantees pivot on ensuring that query cost, recall, and throughput vary smoothly—ideally sublinearly—with selectivity and are robust to filter distribution shifts. The main axes are:

• Selectivity scaling: Solutions such as adaptive-local HNSW (Sehgal et al., 29 Jun 2025), partitioned trees (Jin et al., 3 Jan 2026), and the HSIG hybrid approach (Liang et al., 2024), ensure that query time either remains nearly flat or grows only logarithmically in $1/\sigma$.
• Correlation robustness: Adaptive methods like NaviX react to filter-positive or -negative correlation with the query vector, dynamically choosing exploration to avoid wasted work (Sehgal et al., 29 Jun 2025).
• High-dimensional stability: Penalized filtering (with sufficient $\alpha$) provably eliminates the collapse of distances as $m\to\infty$, in contrast to base vector search (Lakshman et al., 13 Dec 2025).
• Space–time trade-offs: Methods such as SIEVE model index construction, memory, and query cost jointly, ensuring minimal overhead while maximizing predictable throughput (Li et al., 16 Jul 2025).

5. Empirical Characterization of Stability

Empirical studies repeatedly demonstrate that state-of-the-art stable index designs deliver orders of magnitude less variance in latency and throughput across broad selectivity ranges and attribute patterns.

Method	Latency Swing (across selectivity)	Throughput Stdev	Notable Experimental Finding
NaviX (Sehgal et al., 29 Jun 2025)	$\sim$1.5× (1–90%)	$\sim$10 ms	Baselines swing 5–100×; flat latency over all $\sigma$
FCVI-HSW (Heidari et al., 19 Jun 2025)	$<$20% increase under shift	Low	Highest stability under all distribution shifts
Curator (Jin et al., 3 Jan 2026)	Query time nearly flat for low $s$	—	20.9× speedup at $s=10^{-3}$ compared to graph prefilter
SIEVE (Li et al., 16 Jul 2025)	1.5–2.5 ms at 95% recall	Minimal	HNSW varies $<$1 ms (high $s$) to $>$12 ms (low $s$)
HSIG (Liang et al., 2024)	QPS nearly flat (small–large range)	—	Outperforms best-of-three dedicated baselines

These results underscore that stability is not a generic property of a search algorithm, but an emergent result arising from balanced, adaptive index organization and careful algorithmic tuning.

6. Comparative Methodologies and Design Principles

Comparison of methods reveals several core design principles for achieving filtered vector search stability:

• Adaptivity: Per-query and per-region adaptivity (e.g., adaptive-local selectivity in NaviX) is superior to any fixed heuristic.
• Multi-strategy unification: Combining pre-, post-, and hybrid-filtering in a single index (e.g., HSIG) is more effective than relying on a single strategy (Liang et al., 2024).
• Filter-aware index allocation: Partitioning or multi-template index construction (SIEVE, Curator) circumvents the high cost of predicate-unaware global search, especially as selectivity decreases.
• Transformation-based stability: Geometric methods (FCVI) and penalized distances (as in stability theory) provide provable bounds and parameterizable trade-offs between recall, efficiency, and robustness under distributional change.
• Space efficiency: Methods such as SIEVE and Curator minimize proliferation of indexes by workload-aware selection and shared structural components (Li et al., 16 Jul 2025, Jin et al., 3 Jan 2026).

7. Practical Considerations, Limitations, and Open Problems

Although recent advances have achieved marked progress, some limitations and open directions persist:

• Index overhead and dynamism: FCVI and SIEVE incur additional storage for multi-indexing or transformation, though this is often modest compared to naive alternatives.
• Dynamic predicates and multi-attribute queries: Most current systems handle static predicate templates efficiently; dynamic, arbitrary predicate composition remains challenging, though FCVI and Curator make headway via transformation or on-the-fly temporary sub-indexes (Heidari et al., 19 Jun 2025, Jin et al., 3 Jan 2026).
• Parameter selection: Theoretical stability requires careful tuning of hyperparameters (e.g., penalty $\alpha$) guided by empirical filter miss rates and vector distance distributions (Lakshman et al., 13 Dec 2025).
• Generalization to complex predicates: Emerging work targets more complex SQL-like filters, geometric conditions, or learned filter projections (Heidari et al., 19 Jun 2025).
• Diversity of datasets and embeddings: Formal results and empirical studies to date primarily use standard datasets (SIFT1M, GloVe, YFCC, Amazon); adaptation to larger or more heterogeneous datasets is a frontier for future investigation (Heidari et al., 19 Jun 2025).

Filtered vector search stability, as established by recent research, now enables robust and scalable deployment of hybrid filtering and similarity search in production systems. The convergence of adaptive algorithms, hybrid index structures, and theoretical guidance presents a comprehensive framework for predictable, high-performance filtered search across modern, attribute-rich datasets.