subsume

Geometric region embeddings for subsumption, entailment, and logical query answering. Boxes, cones, octagons, Gaussians, hyperbolic intervals, and sheaf networks. Ndarray and Candle backends.

(a) Containment: nested boxes encode taxonomic is-a relationships. (b) Gumbel soft boundary: temperature controls membership sharpness. (c) Octagon: diagonal constraints cut corners for tighter volume bounds.

What it provides

Geometric primitives

Scoring and query answering

Taxonomy and training

Backends

Usage

[dependencies]
subsume = { version = "0.1.8", features = ["ndarray-backend"] }
ndarray = "0.16"

use subsume::ndarray_backend::NdarrayBox;
use subsume::Box as BoxTrait;
use ndarray::array;

// Box A: [0,0,0] to [1,1,1] (general concept)
let premise = NdarrayBox::new(array![0., 0., 0.], array![1., 1., 1.], 1.0)?;

// Box B: [0.2,0.2,0.2] to [0.8,0.8,0.8] (specific, inside A)
let hypothesis = NdarrayBox::new(array![0.2, 0.2, 0.2], array![0.8, 0.8, 0.8], 1.0)?;

// Containment probability: P(B inside A)
let p = premise.containment_prob(&hypothesis, 1.0)?;
assert!(p > 0.9);

Examples

cargo run -p subsume --example containment_hierarchy    # taxonomic is-a relationships with nested boxes
cargo run -p subsume --example gumbel_box_exploration   # Gumbel boxes, soft containment, temperature effects
cargo run -p subsume --example cone_training            # training cone embeddings on a taxonomy
cargo run -p subsume --example box_training             # training box embeddings on a 25-entity taxonomy
cargo run -p subsume --example taxobell_demo            # TaxoBell Gaussian box losses on a mini taxonomy
cargo run -p subsume --example query2box                # Query2Box: multi-hop queries, box intersection, distance scoring
cargo run -p subsume --example octagon_demo             # octagon embeddings: diagonal constraints, containment, volume
cargo run -p subsume --example fuzzy_query              # fuzzy query answering: t-norms, De Morgan duality, rankings
cargo run -p subsume --example el_training              # EL++ box embeddings on a biomedical-style ontology
cargo run -p subsume --features candle-backend --example taxobell_training  # TaxoBell MLP encoder training (Candle)

See examples/README.md for a guide to choosing the right example.

Tests

cargo test -p subsume

Unit, property, and doc tests covering:

• Box geometry: intersection, union, containment, overlap, distance, volume, truncation
• Gumbel boxes: membership probability, temperature edge cases, Bessel volume
• Cones: angular containment, negation closure, aperture bounds
• Octagon: intersection closure, containment, Sutherland-Hodgman volume
• Fuzzy: t-norm/t-conorm commutativity, associativity, De Morgan duality
• Gaussian boxes, EL++ ontology losses, sheaf networks, hyperbolic geometry, quasimetrics
• Training: MRR, Hits@k, NDCG, calibration, negative sampling, AMSGrad

Choosing a geometry

Why Gumbel boxes?

Containment loss under increasing coordinate noise for Gumbel, Gaussian, and hard boxes. Gumbel boxes remain stable at perturbation levels where other formulations fail.

Gumbel boxes model coordinates as Gumbel random variables, creating soft boundaries that provide dense gradients throughout training. Hard boxes create flat regions where gradients vanish; Gumbel boxes solve this local identifiability problem (Dasgupta et al., 2020). As shown above, this also makes containment robust to coordinate noise -- Gumbel containment loss stays near zero even at high perturbation levels where Gaussian boxes fail completely.

Training convergence

25-entity taxonomy learned over 200 epochs. Left: total violation drops 3 orders of magnitude. Right: containment probabilities converge to 1.0 at different rates depending on hierarchy depth. Reproduce: cargo run --example box_training or uv run scripts/plot_training.py.

References

• Vilnis et al. (2018). "Probabilistic Embedding of Knowledge Graphs with Box Lattice Measures"
• Nickel & Kiela (2017). "Poincare Embeddings for Learning Hierarchical Representations"
• Abboud et al. (2020). "BoxE: A Box Embedding Model for Knowledge Base Completion"
• Dasgupta et al. (2020). "Improving Local Identifiability in Probabilistic Box Embeddings"
• Ren et al. (2020). "Query2Box: Reasoning over Knowledge Graphs using Box Embeddings"
• Hansen & Ghrist (2019). "Toward a Spectral Theory of Cellular Sheaves"
• Bodnar et al. (2022). "Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs"
• Zhang & Wang (2021). "ConE: Cone Embeddings for Multi-Hop Reasoning over Knowledge Graphs"
• Chen et al. (2022). "Fuzzy Logic Based Logical Query Answering on Knowledge Graphs"
• Jackermeier et al. (2023). "Dual Box Embeddings for the Description Logic EL++"
• Yang, Chen & Sattler (2024). "TransBox: EL++-closed Ontology Embedding"
• Charpenay & Schockaert (2024). "Capturing Knowledge Graphs and Rules with Octagon Embeddings"
• Huang et al. (2023). "Concept2Box: Joint Geometric Embeddings for Learning Two-View Knowledge Graphs"
• Yang & Chen (2025). "Achieving Hyperbolic-Like Expressiveness with Arbitrary Euclidean Regions"
• Mishra et al. (2026). "TaxoBell: Gaussian Box Embeddings for Self-Supervised Taxonomy Expansion" (WWW '26)

See also

• hyperball -- hyperbolic geometry primitives (direct dependency for Poincare/Lorentz embeddings)

License

MIT OR Apache-2.0