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Neural Spacetimes for DAG Representation Learning
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Proposes Neural SpaceTimes (NSTs), trainable product geometries combining a learnable neural quasi-metric (space) and a learnable neural partial order (multi-dimensional time) to embed weighted DAGs as causal events, with universal embedding guarantees.
Problem
Embedding weighted directed acyclic graphs into continuous spaces that jointly preserve edge directionality (causality) and edge weights (distances), using fixed or learned geometries.
Approach
Decouple a representation into a product of a learnable quasi-metric (neural snowflake-style, on spatial dimensions) and a learnable multi-time-dimension partial order (product order via positive-weight networks), trained end-to-end with an MLP encoder so that the geometry itself is learned rather than fixed as Minkowski or de Sitter.
Key results
- On synthetic DAGs with embedding dimension 2, NST achieves average distortion 1.09 with max 3.18 and directionality 1.0, versus Minkowski's 2.86 average distortion with max 72.66 in the same setting.
- On real-world gene regulatory networks at dimension 2, NST gets distortion 1.00 to 1.06 with directionality 1.00 on In silico, E. coli and S. cerevisiae, while Minkowski has distortions up to 105.42.
- Theory: any k-point DAG embeds into an NST with 1+O(log k) distortion while exactly preserving causal structure; planar Hasse diagrams need only O(log k)+2 spatial and 2 temporal dimensions.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Neural Spacetime instead of Minkowski | DAG embedding · Cornell, embedding dim 2 | distortion: 1.07 ± 0.70 -> 1.00 ± 0.07 | Helped |
| Use Neural Spacetime instead of Minkowski | DAG embedding · Texas, embedding dim 2 | distortion: 1.12 ± 1.73 -> 1.01 ± 0.10 | Helped |
| Use Neural Spacetime instead of De Sitter | DAG embedding · Wisconsin, embedding dim 2 | distortion: 2.06 ± 63.46 -> 1.00 ± 0.10 | Helped |
| Use Neural Spacetime instead of Minkowski | DAG embedding · In silico, embedding dim 2 | distortion: 105.42 ± 4671.85 -> 1.06 ± 0.47 | Helped |
| Use Neural Spacetime instead of Minkowski | DAG embedding · E. coli, embedding dim 2 | distortion: -4.25 ± 149.61 -> 1.02 ± 0.45 | Helped |
| Use Neural Spacetime instead of Minkowski | DAG embedding · S. cerevisiae, embedding dim 2 | distortion: -2.38 ± 173.57 -> 1.05 ± 0.34 | Helped |
| Use Neural Spacetime instead of Minkowski | Synthetic weighted DAG embedding · Synthetic weighted DAGs, embedding dim 2 | average distortion: 2.86 ± 5.22 -> 1.09 ± 0.24 | Helped |
Surprising findings
- Fixed-geometry De Sitter baselines produce wildly unstable distortions (often negative or infinite averages), indicating poor optimization of closed-form spacetimes.
Limitations
- Guarantees apply only to DAGs, not arbitrary digraphs.
- Optimization is easier locally than globally; global ground-truth geodesics are expensive for large graphs.
- Real-world datasets contain directed cycles, so they cannot be embedded perfectly in time.
Experimental setup
Datasets: synthetic weighted DAGs embedded in 2D plane Cornell Texas Wisconsin (WebKB) In silico gene regulatory network E. coli gene regulatory network S. cerevisiae gene regulatory network
Benchmarks: Cornell Texas Wisconsin In silico E. coli S. cerevisiae synthetic weighted DAGs
Metrics: distortion (average and max) directionality accuracy
Baselines: Minkowski spacetime embedding De Sitter spacetime embedding
Builds on: neural snowflake Minkowski spacetime embedding de Sitter spacetime embedding ultrahyperbolic representations
Closest papers by meaning
- Spacetime Representation Learning (ICLR 2023 · similarity 0.89)
- Connecting Neural Models Latent Geometries with Relative Geodesic Representations (NeurIPS 2025 · similarity 0.86)
- Grounding Continuous Representations in Geometry: Equivariant Neural Fields (ICLR 2025 · similarity 0.85)
- Spectro-Riemannian Graph Neural Networks (ICLR 2025 · similarity 0.85)
- Geometry-Aware Similarity Metrics for Neural Representations on Riemannian and Statistical Manifolds (NeurIPS 2026 · similarity 0.84)
- Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models (NeurIPS 2025 · similarity 0.84)
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