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Scalable extraction and visualization of multi-attribute logical and functional dependencies in tabular data

Chaithra Umesh, Arvind Lomrore, Neethu D, Kristian Seegel-Schultz, Saptarshi Bej, Olaf Wolkenhauer

arXiv:2610.08287v1cs.LG

TL;DR

Scalable discovery of multi-attribute LDs remains underexplored, and unified extraction of LDs and FDs is lacking. The paper presents LDTool and HLDTool for dependency extraction, with HLDTool using hypergraph-guided search, and reports matching FDTool’s FDs with lower runtime in high-dimensional feature spaces.

  • Problem

    Scalable extraction of multi-attribute LDs and unified extraction of LDs and FDs remain underexplored as attribute count and dependency order increase.

    Source: §1 · p. 3

  • Method

    LDTool and HLDTool form a unified framework for extracting multi-attribute LDs and FDs, with HLDTool using hypergraph-guided search.

    Source: §1 · p. 4, §5 · p. 22, §6 · p. 24

  • Results

    LDTool recovers the same FDs as FDTool for evaluated dependency layers while requiring less runtime for datasets with larger feature spaces.

    Source: §5 · p. 21, §6 · p. 24

  • Takeaways & Limitations

    The framework targets dependency structures beyond pairwise analysis, including multi-attribute LDs and FDs.

    Source: §5 · p. 22

    The vantage-point tree is a heuristic index rather than an exact one.

    Source: §3.4 · p. 14

Abstract

from arXiv · show

Understanding the structural relationships among attributes in tabular data is fundamental to machine learning and pattern recognition. While functional dependency (FD) discovery has been extensively studied, scalable discovery of logical dependencies (LDs), particularly as the number of attributes and dependency order increase, remains underexplored. These dependencies capture non-deterministic, condition-specific relationships among pairwise or multiple attributes. Furthermore, existing approaches do not provide a unified framework for extracting multi-attribute LDs and FDs. To address these limitations, we propose LDTool and HLDTool for extracting and visualizing multi-attribute LDs and FDs from tabular data. LDTool extends dependency discovery beyond pairwise relationships, while HLDTool enables scalable extraction through hypergraph-guided search-space reduction. Experiments on three simulated and eleven real-world datasets demonstrate that the proposed framework extracts meaningful LDs and FDs while improving scalability. LDTool recovers the same FDs as existing FD discovery methods with lower runtime in high-dimensional feature spaces, whereas HLDTool enables dependency discovery in datasets with hundreds of features. The proposed framework provides interpretable visualizations of dependency structures and supports applications in exploratory data analysis and the quantitative evaluation of synthetic tabular data.

1. Introduction

The paper targets the limited extraction and interpretation of multi-attribute LDs alongside FDs, especially as feature dimensionality and dependency order grow. It presents LDTool and HLDTool for joint extraction, scalable search, and visualization, and reports lower runtime than FDTool while recovering the same FDs in evaluated layers.

1. Introduction

  • Multi-attribute LD extraction remains largely unexplored, and existing methods do not jointly extract multi-attribute LDs and FDs.The paper also notes that practitioners commonly rely on pairwise summaries and that higher-order relationships can be missed by pairwise analysis alone.Source: §1 · pp. 3–4
  • Dependency search grows rapidly with attribute count and dependency order, making higher-order extraction computationally expensive for high-dimensional data.Lattice-based discovery may involve up to 2^n attribute subsets.Source: §1 · p. 4
  • LDTool jointly identifies LDs and FDs across user-specified attribute-lattice levels, representing deterministic and rule-based relationships.The framework is intended to provide a more comprehensive view than FD-focused approaches.Source: §1 · pp. 4–5
  • HLDTool groups strongly related attributes for local dependency discovery, while a VP-tree approximation accelerates extraction in high-dimensional spaces.The authors say the search-space reduction preserves meaningful multi-attribute relationships.Source: §1 · p. 5
  • Force-directed layouts, dendrograms, and hypergraphs visualize multi-attribute dependencies for interpretable exploration beyond pairwise correlation analysis.The resulting structures are also proposed as a basis for comparing structural consistency in real and synthetic tabular data.Source: §1 · pp. 5–6
  • LDTool recovers the same multi-attribute FDs as FDTool up to three dependency layers while reducing runtime, especially on high-dimensional datasets.Experiments cover three simulated and eleven real-world datasets, and the authors report that multi-view visualizations reveal interpretable dependency structures.Source: §1 · pp. 5–6

2. Related research

Prior work spans classical FD mining, relaxed dependency semantics, and graph-based visualization, but the cited approaches do not provide a unified framework for scalable extraction and visualization of multi-attribute LDs and FDs.

2. Related research

  • Classical FD systems established efficient discovery strategies, including lattice traversal, stripped partitions, and pruning.The cited tools include TANE, Dep-Miner, FUN, FD_Mine, and FDTool.Source: §2 · pp. 6–8
  • Relaxed FDs, conditional dependencies, inclusion dependencies, and approximate or statistical methods broaden dependency discovery beyond exact classical FDs.The paper distinguishes its LD semantics from these dependency classes.Source: §2 · pp. 6–7
  • The paper characterizes its LDs as directional relationships where the left-hand side constrains possible right-hand-side values without functionally determining them.Their strength is quantified through the Q-function.Source: §2 · p. 7
  • Prior approximate and statistical approaches largely addressed noisy or approximate global relationships and pairwise associations rather than multi-attribute LDs.The authors also note that FDTool experiments found attribute count affects runtime and memory more than record count.Source: §2 · p. 7
  • Graph-based dependency visualizations support structural exploration, but cited approaches were not designed for unified FD and LD discovery or higher-order dependencies.The paper says earlier work also lacks a unified framework for scalable extraction, quantification, and visualization of multi-attribute LDs and FDs.Source: §2 · p. 7
  • Dependency-aware synthetic-data methods use learned or predefined structures to guide generation, rather than directly discovering, quantifying, and visualizing multi-attribute LDs and FDs.Examples include Kamino, CuTS, SAGE, PRISM, MoE-T, and LLM-TabFlow.Source: §2 · p. 8

constrain the generation process, rather than directly discovering, quantifying,

Multi-attribute LD extraction remains underexplored, and existing approaches lack a common framework for extracting both LDs and FDs while making complex dependency structures interpretable. The paper introduces a unified framework with scalable search and multiple visual representations.

constrain the generation process, rather than directly discovering, quantifying,

  • Existing methods largely focus on specific dependency classes, while methods jointly extracting multi-attribute LDs and FDs within one framework are lacking.The paper also identifies difficulty interpreting individual-rule outputs as dependency number and order increase.Source: constrain the generation process,… · p. 8
  • The proposed framework combines multi-attribute LD and FD extraction with hypergraph-guided search and interpretable visualizations.Its visual forms include force-directed graphs, dendrograms, and hypergraphs.Source: constrain the generation process,… · pp. 8–9

3. Methodology

The methodology scores dependencies with a bounded Q-function and uses LDTool and HLDTool to extract multi-attribute LDs and FDs, scaling the search through hypergraph-guided grouping and approximate nearest-neighbor search. It also describes a vantage-point-tree heuristic for approximating the Q-matrix and the datasets, validation procedures, preprocessing, and parameter settings used in evaluation.

3.1. Preliminaries: the Q-function

  • The Q-function scores a rule using distinct observed LHS–RHS combinations, not raw row counts; q = 0 denotes an FD, while 0 < q < 1 denotes an LD.Source: §3.1 · pp. 9–10

3.2. Unified multi-attribute LD and FD extraction (LDTool)

  • LDTool discovers LDs and FDs together for attribute sets up to a chosen maximum layer, then filters redundant rules using dependency-specific minimality criteria.For LDs, a higher-order rule is retained only if its strength improves by at least τLD over every smaller LD with the same RHS; the method uses τLD = 0.2.Source: §3.2 · pp. 10–12

3.3. Hypergraph-guided scalable dependency discovery (HLDTool)

  • HLDTool groups strongly related attributes into hyperedges and runs LDTool within each group, reducing candidate subsets while keeping related attributes together.Pairwise q-scores identify candidates below a user-defined threshold; the top-k candidates form each hyperedge, with k controlling the coverage–efficiency trade-off.Source: §3.3 · p. 13

3.4. Fast approximation of the Q-matrix using a vantage-point tree

  • To avoid the O(n^2) pairwise Q-matrix bottleneck at very high dimensionality, the method approximates nearest-neighbor search over attributes using a vantage-point tree.Source: §3.4 · pp. 13–14
  • Because q is asymmetric and lacks the triangle inequality, the VP tree is a heuristic rather than an exact metric index, validated empirically against brute-force computation.The comparison is reported in Table 4; the supplied passages do not provide its results.Source: §3.4 · p. 14
  • A VP tree partitions attributes by pivot-distance medians, then branch-and-bound k-NN search retains each attribute’s k strongest dependencies in a sparse asymmetric approximation of Q, avoiding all n^2 pairs.The search maintains a radius to the current k-th best match and prunes subtrees that cannot contain a closer neighbor.Source: §3.4 · pp. 15–16

3.5. Experimental Setup

  • Evaluation covers 14 datasets (3 simulated, 11 real-world), spanning 8–642 attributes and approximately 100–45,000 instances across diverse domains and dimensionalities.Table 1 summarizes dataset characteristics and extracted LD and FD counts.Source: §3.5 · p. 16
  • Before extraction, missing values are removed and categorical attributes numerically encoded; METABRIC expression is quantile-binned, mutations binarized, and only first-layer dependencies extracted.These METABRIC-specific restrictions address its high feature dimensionality.Source: §3.5 · p. 16
  • Extracted LDs are checked on the complete dataset against the adopted criterion, FDs against FDTool, and simulated dependencies provide partial ground truth.The simulated data may also contain valid dependencies from transitive relationships or combinations of attributes.Source: §3.5 · p. 16
  • LDTool uses τs = 0.8 (q ≤ 0.2) and τLD = 0.2 by default to extract FDs and strong LDs while limiting marginal-strength improvements.The stated rationale is that smaller τLD retains many additional LDs with marginal gains, while larger values do not further reduce retained LDs.Source: §3.5 · p. 16
  • HLDTool constructs hypergraphs with τH = 0.2 and maximum hyperedge size 20, using the threshold for consistent comparisons across dataset dimensions.The influence of τH is evaluated in Section 4.Source: §3.5 · p. 16

4. Results

The results describe interpretable multi-attribute dependency extraction with LDTool and the use of HLDTool and VP-tree approximation to make discovery more practical as dimensionality increases, including tradeoffs between dependency coverage and computational cost. They also explain the choice of τLD = 0.2, which did not discard additional informative dependencies.

LDTool extracts meaningful multi-attribute logical dependencies from tabular datasets

  • In the student dataset, LDTool identified a dependency between field of study and stress level, with Business or STEM stress consistently no higher than 8/10; higher late-night usage corresponded to shorter sleep.Source: Figure 2 · p. 17, §4 · p. 18
  • In the house-purchase dataset, city relates to salary, while crime and legal cases on a property relate to purchase decisions.Abu Dhabi customers consistently had higher salary ranges; properties with legal cases were rejected, and so were properties with at least three reported crime cases even without a legal case.Source: Figure 2 · p. 17, §4 · p. 18
  • LDTool extracted all designed dependencies in Simulated_data_3 and Simulated_data_7, and 10 of 11 in Simulated_data_4; the remaining LD was excluded because its q-score failed the selected strong-LD criterion (q ≤0.2).Source: §4 · p. 18

HLDTool enables scalable discovery of multi-attribute LDs and FDs in high-dimensional feature spaces

  • In US Census, HLDTool with maximum hyperedge size 20 recovered 1,306 LDs and 8 FDs in approximately one minute, versus LDTool’s 224,767 LDs and 20 FDs in 306 minutes.For FD-only extraction, LDTool recovered the same 20 FDs in 44 minutes, compared with 60 minutes for FDTool.Source: §4 · p. 19
  • On TIC Insurance and METABRIC, HLDTool found thousands of higher-order LDs and FDs within minutes, where exhaustive third-layer extraction takes days or weeks.The complete dependency set could not be established for these datasets because exhaustive extraction was infeasible.Source: §4 · p. 20

VP-tree-based neighborhood search accelerates first-layer dependency extraction

  • On METABRIC, VP-tree recovered more than 95% of dependencies in seconds, versus approximately seven minutes for exhaustive computation.Across datasets, VP-tree reduced runtime while preserving most first-layer dependencies; low- and medium-dimensional results closely matched exhaustive computation.Source: §4 · p. 21

Sensitivity analysis of LD extraction parameters

  • A strength threshold τs = 0.8 (q ≤ 0.2) retained strong LDs while capturing all FDs (q = 0); lower τs increased weaker LDs, while τLD < 0.2 often yielded thousands of additional dependencies.Source: §4 · p. 21

4. Results

  • τLD = 0.2 was adopted because higher values did not further reduce retained LDs, while the chosen threshold preserved informative dependencies.The passages also report marginal improvements in dependency strength.Source: §4 · p. 21

5. Discussion

LDTool matches FDTool’s FD counts while reducing runtime in higher-dimensional datasets, and the discussion illustrates how multi-attribute dependencies and complementary visualizations enrich structural analysis. It also identifies limits on completeness and on generalizing the findings beyond the evaluated settings.

5. Discussion

  • LDTool extracted first-layer FDs in under two minutes, while FDTool failed to extract a single FD from the high-dimensional dataset.Source: §5 · p. 22
  • LDTool extracts LDs and FDs simultaneously, extending dependency analysis beyond pairwise relationships to capture interactions among multiple attributes.Source: §5 · p. 22
  • LDTool identified the same number of FDs as FDTool across datasets, with lower FD-only runtime particularly for datasets with more features and comparable runtime on smaller datasets.Table 5 compares extraction up to three layers, except METABRIC, which is restricted to the first layer.Source: §5 · p. 21, Table 5 · p. 22
  • In the global house purchase dataset, the purchase decision depends jointly on legal cases and reported crime counts, a relationship neither attribute alone fully characterizes.Source: §5 · p. 22
  • Force-directed, dendrogram, and hypergraph views form coherent dependency groups consistent with the manually constructed FDTool structure.The figure also contrasts these views with the manually constructed visualization.Source: §5 · p. 23
  • Completeness is difficult to assess because exhaustive higher-order extraction is computationally infeasible, and HLDTool’s pairwise-dependency hyperedges may miss dependencies involving weakly related features.Source: §5 · p. 24
  • Whether higher-order LDs and FDs reveal interpretable relationships in large-scale biological and biomedical data beyond METABRIC, or help assess synthetic-data structural fidelity, remains open.Source: §5 · p. 24

6. Conclusion

The conclusion presents LDTool and HLDTool as a unified framework for extracting and visualizing multi-attribute logical and functional dependencies, with scalable search and interpretable dependency representations. It reports extraction results while noting limits to completeness assessment in high-dimensional data and identifying possible applications and future research.

6. Conclusion

  • LDTool supports exhaustive LD and FD discovery up to a user-specified layer, while HLDTool uses hypergraph-guided search and VP-tree approximation to improve scalability and accelerate pairwise discovery.The framework also includes graph and dendrogram visualizations to support interpretation of dependency structures.Source: §6 · pp. 24–25
  • Across simulated and real-world datasets, the framework extracts multi-attribute LDs alongside FDs, and LDTool recovers the same FDs as FDTool for evaluated dependency layers with less runtime on larger feature spaces.The authors describe the approach as practical and scalable for studying multi-attribute dependencies in tabular data.Source: §6 · pp. 24–25
  • The framework's graph and hypergraph representations support interpretation of complex dependency structures.Source: §6 · p. 25
  • HLDTool restricts multi-attribute search using pairwise dependency information, so dependencies among features with weak pairwise relationships may be missed.The authors say assessing completeness remains challenging in high-dimensional settings where exhaustive extraction is infeasible.Source: §6 · p. 25
  • The authors identify high-dimensional biological and biomedical datasets as potential application areas and call for strategies to assess dependency coverage when exhaustive extraction is infeasible.They also propose examining whether LDs and FDs complement structural-fidelity measures for synthetic tabular data and whether incorporating them into generative models can improve preservation of multi-attribute relationships.Source: §6 · p. 25

CRediT authorship contribution statement

The contribution statement assigns experiments, analysis, implementation, literature review, mathematical review, and manuscript supervision across the named authors.

CRediT authorship contribution statement

Availability of code and results

The authors state that detailed experiment Jupyter notebooks are available on GitHub to support transparency, reusability, and reproducibility.

Availability of code and results

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process

The authors report using ChatGPT to generate a reference image for visualizing extracted dependencies, while manually creating the final figure. They state that AI tools were not used for manuscript text, analyses, or scientific conclusions.

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process

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