Region embeddings for entailment and set containment.
(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.
Boxes, Gumbel boxes, cones, octagons, Gaussians, and hyperbolic intervals. Each
implements containment probability, volume, intersection, and distance. CPU
backend via ndarray, GPU via candle (features = ["candle-backend"] or ["cuda"]).
Scoring: Query2Box distance, fuzzy t-norms for logical queries, EL++ ontology losses (Box2EL/TransBox). Training: CPU trainer with analytical gradients or GPU trainer with AdamW autograd. Evaluation: MRR, Hits@k, Mean Rank (filtered).
[dependencies]
subsume = { version = "0.9", features = ["ndarray-backend"] }
ndarray = "0.16"use subsume::ndarray_backend::NdarrayBox;
// Renamed import avoids shadowing std::boxed::Box
use subsume::Box as BoxRegion;
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)?;
assert!(p > 0.9);use subsume::{BoxEmbeddingTrainer, TrainingConfig, Dataset};
use subsume::dataset::load_dataset;
use std::path::Path;
let dataset = load_dataset(Path::new("data/wn18rr"))?;
let interned = dataset.into_interned();
let train: Vec<_> = interned.train.iter().map(|t| (t.head, t.relation, t.tail)).collect();
let config = TrainingConfig { learning_rate: 0.01, epochs: 50, ..Default::default() };
let mut trainer = BoxEmbeddingTrainer::new(config, 32);
let result = trainer.fit(&train, None, None)?;
println!("MRR: {:.3}", result.final_results.mrr);use subsume::CandleBoxTrainer;
use candle_core::Device;
let device = Device::cuda_if_available(0).unwrap_or(Device::Cpu);
let trainer = CandleBoxTrainer::new(num_entities, num_relations, 200, 10.0, &device)?
.with_inside_weight(0.02) // BoxE-style center attraction
.with_vol_reg(0.0001); // prevent trivial solution
let losses = trainer.fit(&train_triples, 500, 0.001, 512, 9.0, 128, 1.0)?;
let (mrr, h1, h3, h10, mr) = trainer.evaluate(&test_triples, &all_triples)?;pip install subsumerimport subsumer
triples = [("animal", "hypernym", "dog"), ("animal", "hypernym", "cat"), ...]
config = subsumer.TrainingConfig(dim=32, epochs=50, learning_rate=0.01)
trainer, ids = subsumer.BoxEmbeddingTrainer.from_triples(triples, config=config)
result = trainer.fit(ids)
print(f"MRR: {result['mrr']:.3f}")Triple convention: head box contains tail box. For datasets where triples
are (child, hypernym, parent), pass reverse=True to from_triples or
load_dataset.
cargo run -p subsume --example box_training # train box embeddings on a 25-entity taxonomy
cargo run -p subsume --example dataset_training --release # full pipeline: WN18RR data, train, evaluate
cargo run -p subsume --example el_training # EL++ ontology embedding18 examples total covering all geometries, training modes, and query types.
See examples/README.md for the full list.
cargo test -p subsumeUnit, 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
- Training: MRR, Hits@k, Mean Rank, negative sampling (uniform, Bernoulli), AMSGrad
| Geometry | When to use it | ¬? | Key tradeoff |
|---|---|---|---|
| NdarrayBox / NdarrayGumbelBox | Containment hierarchies, each dimension independent | No | Simple, fast; Gumbel adds dense gradients where hard boxes have zero gradient |
| Cone | Multi-hop queries requiring negation (FOL with ¬) | Yes | Closed under complement; angular parameterization harder to initialize |
| Octagon | Rule-aware KG completion; tighter containment than boxes | No | Diagonal constraints cut box corners; more parameters per entity |
| Gaussian | Taxonomy expansion with uncertainty (TaxoBell) | No | KL = asymmetric containment; Bhattacharyya = symmetric overlap |
| Hyperbolic | Tree-like hierarchies with exponential branching | No | Low-dim capacity; numerical care near Poincare ball boundary |
Point embeddings (TransE, RotatE, ComplEx) represent entities as vectors. They work well for link prediction -- RotatE hits 0.476 MRR on WN18RR, BoxE hits 0.451. For standard triple scoring, points are simpler and equally accurate.
Regions become necessary when the task requires structure that points cannot encode. The core operation is containment probability:
If B fits inside A, containment_prob).
| What you need | Points | Regions |
|---|---|---|
| Containment (A ⊆ B) | No -- points have no interior | Box nesting = subsumption |
| Volume = generality | No -- points have no size | Large box = broad concept |
| Intersection (A ∧ B) | No set operations | Box ∩ Box = another box |
| Negation (¬A) | No complement | Cone complement = another cone |
| Uncertainty per dimension | No | Gaussian sigma |
Three tasks where point embeddings structurally fail:
-
Ontology completion (EL++): "Dog is-a Animal" requires representing one concept's extension as a subset of another's. Points have no containment. Box2EL, TransBox, and DELE use boxes for this and outperform point baselines on Gene Ontology, GALEN, and Anatomy.
-
Logical query answering (∧, ∨, ¬): multi-hop KG queries with conjunction, disjunction, and negation need set operations. ConE handles all three (MRR 52.9 on FB15k EPFO+negation queries vs Query2Box's 41.0 and BetaE's 44.6). Points cannot attempt negation queries at all.
-
Taxonomy expansion: inserting a new concept at the right depth requires knowing both what it is (similarity) and how general it is (volume). TaxoBell uses Gaussian boxes where KL divergence gives asymmetric parent-child containment for free.
If your task is link prediction or entity similarity, use RotatE. If you need containment, set operations, or volume, you need regions.
See docs/SUBSUMPTION_HISTORY.md for the research
history of geometric subsumption embeddings, from hard boxes through Gumbel, cones, and beyond.
(a) Membership probability at a box boundary: hard boxes have a discontinuous step, Gumbel boxes have smooth sigmoids controlled by temperature. (b) Gradient magnitude: hard boxes produce zero gradient everywhere except the exact boundary (gray regions), while Gumbel boxes provide gradients throughout the space.
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). Lower temperature (small beta) gives crisper boundaries with sharper gradients; higher temperature gives broader gradients that reach further from the boundary but sacrifice containment precision.
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.
BoxEmbeddingTrainer::export_embeddings() returns flat f32 vectors suitable for
safetensors, numpy (via reshape), and vector databases:
let (ids, mins, maxs) = trainer.export_embeddings();
// mins/maxs are flat Vec<f32> of length n_entities * dim
// Reshape to (n_entities, dim) for numpy/safetensorsCheckpoint save/load via serde:
let json = serde_json::to_string(&trainer)?;
let restored: BoxEmbeddingTrainer = serde_json::from_str(&json)?;Convert from petgraph (when petgraph feature is enabled):
use subsume::petgraph_adapter::from_graph;
let dataset = from_graph(&my_digraph);Convert from polars (no dependency needed, user-side code):
use subsume::dataset::Triple;
let triples: Vec<Triple> = df.column("head")?.str()?
.into_iter()
.zip(df.column("relation")?.str()?)
.zip(df.column("tail")?.str()?)
.filter_map(|((h, r), t)| Some(Triple::new(h?, r?, t?)))
.collect();
let dataset = Dataset::new(triples, vec![], vec![]);Per-normal-form results on Box2EL benchmark datasets (Jackermeier et al., 2023), evaluated by center L2 distance ranking (matching Box2EL protocol). Default hyperparameters: dim=200, epochs=300, lr=0.01, margin=0.15, batch=512. Bold marks the best H@1 for each (dataset, NF) pair.
| Dataset | NF type | subsume H@1 | subsume H@10 | Box2EL H@1 | Box2EL H@10 |
|---|---|---|---|---|---|
| GALEN (23K) | NF1: C1 ⊓ C2 ⊑ D | 0.000 | 0.000 | 0.03 | 0.30 |
| GALEN | NF2: C ⊑ D | 0.036 | 0.158 | 0.06 | 0.15 |
| GALEN | NF3: C ⊑ ∃r.D | 0.113 | 0.299 | 0.08 | 0.19 |
| GALEN | NF4: ∃r.C ⊑ D | 0.000 | 0.001 | 0.00 | 0.06 |
| GO (46K) | NF1 | 0.000 | 0.000 | 0.03 | 0.17 |
| GO | NF2 | 0.010 | 0.096 | 0.18 | 0.58 |
| GO | NF3 | 0.006 | 0.033 | 0.00 | 0.018 |
| GO | NF4 | 0.000 | 0.000 | 0.00 | 0.37 |
| ANATOMY (106K) | NF1 | 0.000 | 0.005 | 0.07 | 0.34 |
| ANATOMY | NF2 | 0.001 | 0.011 | 0.16 | 0.41 |
| ANATOMY | NF3 | 0.085 | 0.115 | 0.021 | 0.056 |
| ANATOMY | NF4 | 0.000 | 0.005 | 0.00 | 0.05 |
subsume wins NF3 (existential) on GALEN and ANATOMY; competitive on GO NF3.
Box2EL wins NF1, NF2, and NF4 on most datasets.
NF1 requires hyperparameter tuning beyond defaults for competitive results.
Results from CandleElTrainer with Box2EL-style bump translations,
inclusion loss, and dual-direction NF3 negative sampling.
NF4 negative sampling can be controlled via nf4_neg_weight (0.0 disables).
Reproduce: BACKEND=candle cargo run --features candle-backend --example el_benchmark --release -- data/GALEN
The CandleBoxTrainer supports CPU, CUDA, and Metal via the candle backend:
# CPU
cargo run --features candle-backend --example wn18rr_candle --release
# CUDA GPU
cargo run --features cuda --example wn18rr_candle --releaseConfigure via environment variables:
DIM=200 EPOCHS=500 LR=0.001 NEG=128 BATCH=512 MARGIN=9.0 \
ADV_TEMP=1.0 INSIDE_W=0.02 VOL_REG=0.0001 BOUNDS_EVERY=50 \
cargo run --features cuda --example wn18rr_candle --releaseSee examples/README.md for all available examples.
- Nickel & Kiela (2017). "Poincare Embeddings for Learning Hierarchical Representations"
- Vilnis et al. (2018). "Probabilistic Embedding of Knowledge Graphs with Box Lattice Measures"
- Li et al. (2019). "Smoothing the Geometry of Probabilistic Box Embeddings" (ICLR 2019)
- Sun et al. (2019). "RotatE: Knowledge Graph Embedding by Relational Rotation in Complex Space" (self-adversarial negative sampling)
- 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"
- Boratko et al. (2021). "Box Embeddings: An open-source library for representation learning using geometric structures" (EMNLP Demo)
- Chen et al. (2021). "Probabilistic Box Embeddings for Uncertain Knowledge Graph Reasoning" (BEUrRE, ACL 2021)
- Gebhart, Hansen & Schrater (2021). "Knowledge Sheaves: A Sheaf-Theoretic Framework for Knowledge Graph Embedding"
- 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"
- Bourgaux et al. (2024). "Knowledge Base Embeddings: Semantics and Theoretical Properties" (KR 2024)
- Charpenay & Schockaert (2024). "Capturing Knowledge Graphs and Rules with Octagon Embeddings"
- Lacerda et al. (2024). "Strong Faithfulness for ELH Ontology Embeddings" (TGDK 2024)
- Huang et al. (2023). "Concept2Box: Joint Geometric Embeddings for Learning Two-View Knowledge Graphs"
- Mashkova et al. (2024). "DELE: Deductive EL++ Embeddings for Knowledge Base Completion"
- 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)
MIT OR Apache-2.0