Vector-Relational Database Queries

• Vector-relational database queries are hybrid models that combine high-dimensional vector search with traditional attribute filtering to enable complex semantic and analogical analytics.
• They employ integrated operators and advanced indexing techniques, such as HNSW and IVF, to support efficient k-NN, range-based, and join operations.
• These queries power applications in retrieval-augmented generation, recommendation systems, and cognitive intelligence, enhancing multi-modal knowledge discovery.

Vector-relational database queries integrate high-dimensional vector search—typically powered by learned embeddings from text, images, or other unstructured sources—with traditional relational (attribute-based) query processing. This hybrid paradigm enables sophisticated analytics on diverse data sources by supporting similarity, semantic, and analogical queries within the declarative framework of relational algebra and SQL. Such capabilities have become central in modern systems due to widespread use cases in retrieval-augmented generation, recommendation, cognitive intelligence, and multi-modal knowledge discovery.

1. Definitions, Data Models, and Query Formalism

A vector-relational (or "hybrid") query processes a dataset in which each tuple consists of both standard relational attributes and one or more dense vector fields. Let the schema for a prototypical hybrid relation be: $$\text{Items}(\text{ID}, \text{Attr}_1, ..., \text{Attr}_m, \mathbf{v} \in \mathbb{R}^d)$$ where $\mathbf{v}$ is a $d$-dimensional embedding vector. A typical k-nearest neighbor (k-NN) query with relational filters can be formalized as: $$R = \mathrm{Top}_k\left(\left\{ t \in \text{Items} \mid P(t.\text{Attr}_1, ..., t.\text{Attr}_m) \wedge f(t.\mathbf{v}, \mathbf{q}) \geq \theta \right\}\right)$$ where:

• $P(\cdot)$ is a predicate over (possibly multiple) scalar or categorical attributes,
• $f(\cdot, \cdot)$ is a similarity function (e.g., cosine similarity, Euclidean distance),
• $\theta$ is a similarity or distance threshold,
• $\mathbf{q}$ is a query embedding.

Hybrid queries extend naturally to approximate k-NN (ANN), range, multi-modal, and analogical queries, as well as joins based both on attribute values and vector similarity (Pan et al., 2023, Ye et al., 31 Oct 2025, Sanca et al., 2023).

2. Query Types and Supported Semantics

Hybrid queries are categorized according to the manner in which vector and relational predicates interact (Pan et al., 2023, Ma et al., 9 Jan 2025, Ye et al., 31 Oct 2025):

• Non-predicated k-NN/ANN: Retrieve k closest items based purely on their vector embeddings.
• Range-based Hybrid Queries: Select items within a specified distance threshold that also satisfy given attribute predicates.
• Single-stage (Integrated) Hybrid Search: Evaluate relational predicates during traversal of vector indexes (e.g., in HNSW graph search) rather than only as pre/post-filters.
• Analogical and Cognitive Queries: Express vector arithmetic constraints (e.g., "A is to B as C is to ?") leveraging operations such as $v_B - v_A + v_C$ and support for semantic operator UDFs in SQL (Bordawekar et al., 2016, Bordawekar et al., 2017, Bandyopadhyay et al., 2020).
• Join and Multi-table Hybrid Queries: Compose relational and vector similarity join operators, most prominently via context-enhanced joins (E-joins) (Sanca et al., 2023).

SQL and SQL-like syntax extensions support all modalities:

SELECT * FROM Items WHERE category='A' ORDER BY cosine_similarity(v, q) DESC LIMIT k;

or with integrated search:

SELECT * FROM Items INTEGRATED_SEARCH(v,q) FILTER (category='A') LIMIT k;

3. Algebraic and Operator Frameworks

Vector-relational queries are supported by several algebraic and operator-level generalizations:

• E-joins (Context-Enhanced Joins): Extend relational algebra with embedding-aware join predicates, formally

$$R \; \underset{E}{\Join}_{\langle f,g,\theta \rangle} \; S = \{ (r,s) \in R \times S \mid \mathrm{sim}(f(r.x), g(s.y)) \geq \theta \}$$

supporting first-class embedding operators $E_{x}^\mathcal{M}(R)$ that append a vector column via model $\mathcal{M}$ (Sanca et al., 2023).

• Composable Embedding Operators: Operators for vector extraction, normalization, and model application can be composed with selection, projection, and join via defined algebraic rewrites ensuring logical and physical optimizability.
• Analogy and Semantic Operators: User-defined functions (UDFs) such as cosine_similarity, analogyUDF, and proximityMax enable semantic, analogical, and clustering queries in SQL and serve as atomic operators in execution plans (Bordawekar et al., 2016, Bordawekar et al., 2017, Bandyopadhyay et al., 2020).

Operator composition and rewrite rules facilitate cost-based optimization, including selection/projection pushdown and efficient physical realization (e.g., via vectorized or index-assisted join execution).

4. Execution Strategies and System Architectures

Efficient execution of vector-relational queries depends on the interplay between selectivity, data layout, index support, and hardware utilization. The space of strategies branches into several principal approaches (Pan et al., 2023, Sanca et al., 2024, Ye et al., 31 Oct 2025, Ma et al., 9 Jan 2025):

Access Paths

• Scan-based (Full/Efficient): Best for low selectivity; exploits SIMD, batching (N:N tensor computation), and data layout to realize high-throughput scans (Sanca et al., 2024).
• Index-based (ANN): Applies for high-selectivity or large data; leverages structures such as HNSW, IVF, or PQ; filters may be pre-applied, post-applied, or integrated within index traversal ("visit-first" hybrid search) (Pan et al., 2023, Ye et al., 31 Oct 2025, Ma et al., 9 Jan 2025).
• Clustered Hybrid (Compass): Combines proximity-graph on all vectors, IVF clustering, and per-cluster B⁺-tree(s) for relational attributes. Candidate generation and filtering are coordinated through a shared queue, using adaptive expansion as dictated by the attribute selectivity and neighborhood connectivity (Ye et al., 31 Oct 2025).
• Native Integration & Plan Optimization (CHASE): Classifies logical plans into hybrid query templates (VKNN-SF, DR-SF, entity joins), applies semantic and physical rewrites to insert ANN-aware physical operators, and uses MLIR–LLVM-based compilation for direct, branch-free execution (Ma et al., 9 Jan 2025).

Hardware and Index Optimizations

• SIMD and Batch Linear Algebra: Matrix-matrix BLAS (GEMM) for batched k-NN, leading to one to two orders of magnitude speedup versus record-wise scans (Bordawekar et al., 2016, Sanca et al., 2024).
• Quantization and Compression: Product quantization, IVFADC, and LSH reduce storage and accelerates computations with bounded accuracy loss (Pan et al., 2023).
• MPP-Native Storage and Index Co-partitioning: In distributed graph/vector systems (e.g., TigerVector), vector indexes are built per graph partition, with distributed query execution and per-segment HNSW traversal (Liu et al., 20 Jan 2025).

Selectivity-dependent Plan Selection

• Scan–Probe Cross-Over: There exists a specific selectivity $s^*$ (analytically derived) at which scan-based access becomes less efficient than index-based; this threshold depends on $d$, hardware parallelism, index quality/overhead, and query concurrency (Sanca et al., 2024).

5. Storage, Indexing, and Data Layout

Efficient vector-relational query execution is grounded in multi-modal data storage and indexing:

• Embedded Attribute Types: Systems implement first-class vector ("embedding") columns with explicit metadata: dimension, model, metric, index parameters (e.g., HNSW) (Liu et al., 20 Jan 2025).
• Per-attribute and Per-cluster Indexing: Vector indexes (HNSW, IVF) are complemented by per-cluster or global B⁺-trees on relational attributes to accelerate hybrid query filtering (Ye et al., 31 Oct 2025).
• Compact and SIMD-aligned Vector Storage: Dense, columnar layout—crucial for block-wise vectorized computations and for compatibility with high-performance BLAS routines (Sanca et al., 2023).
• Transactional and MVCC-aware Update Handling: Vector indexes incorporate change logs and background vacuuming for transactional consistency with base relation partitions in distributed settings (Liu et al., 20 Jan 2025).

6. Theoretical and Probabilistic Extensions

Recent efforts connect vector-relational queries with probabilistic databases and algebraic frameworks:

• Probabilistic Embedding Models: Embedding-based scores are interpreted probabilistically under the tuple-independent PDB semantics, with marginal and conjunctive query evaluation possible via tractable models such as TractOR (Friedman et al., 2020).
• Algebraic Foundations and Modules: Relational operations (union, intersection, join) are modeled as (multi-)linear maps over modules and polysets, enabling worst-case optimal execution for cyclic joins and symbolic representation of infinite or parameterized relations (Henglein et al., 2022).

7. Open Challenges and Implications

Several practical and theoretical gaps remain:

• Plan Enumeration and Cost Model Fidelity: Precise cost models for hybrid operators, in particular for integrated scan strategies, are required for robust plan selection (Pan et al., 2023).
• Cardinality Estimation for Similarity Predicates: Estimating selectivity and cardinality under high-dimensional similarity is unresolved (Sanca et al., 2023).
• Model Management and Integration: Unified frameworks for model training, online update, and embedding versioning within DBMS remain under active development (Ye et al., 31 Oct 2025, Liu et al., 20 Jan 2025).
• Memory and Scalability: Pure-DRAM vector indexes are restrictive; efficient hybrid disk-memory schemes (e.g., IVF+PQ, ILL-trees) are needed for terabyte-scale systems (Liu et al., 20 Jan 2025).
• Privacy and Security: High-dimensional indexing and search raise non-trivial issues for access control, query privacy, and secure computation (Pan et al., 2023).

A plausible implication is that further convergence of logical/physical database design, hardware-aware execution, and machine learning model integration will continue to shape the future of high-performance, explainable, and multi-modal analytics engines.