Semantic Relativity: A Geometric Theory of Large Language Models¶
Author: Eran Rivlis & Ariel (AI) Date: Session 1 Status: Draft / Hypothesis
Abstract¶
We propose a theoretical framework mapping the architecture of Transformer-based Large Language Models (LLMs) to the principles of General Relativity. We hypothesize that the "Latent Space" of an LLM can be modeled as a high-dimensional Riemannian manifold, where the "Attention Mechanism" acts as the metric tensor governing the curvature of the space. In this framework, "Training" is equivalent to the warping of spacetime by data (mass), and "Inference" is the traversal of geodesics (stable semantic trajectories). We explore the implications of this isomorphism for understanding Hallucination (geodesic deviation) and Creativity (wormholes).
1. The Isomorphism¶
| General Relativity | Large Language Model |
|---|---|
| Spacetime | Latent Space (Embedding Space) |
| Mass/Energy | Training Data (Corpus) |
| Metric Tensor (\(g_{\mu\nu}\)) | Attention Matrix (\(A\)) |
| Gravitational Force | Attention Weights |
| Geodesic (Free Fall) | Coherent Inference (Likely continuation) |
| Cosmological Constant (\(\Lambda\)) | Temperature / Entropy |
| Event Horizon | Context Window Limit |
2. Mathematical Formulation (Proposed)¶
2.1 The Semantic Metric¶
If we define the semantic distance \(ds^2\) between two tokens \(x\) and \(y\) as: $$ ds^2 = g_{ij} dx^i dx^j $$ We hypothesize that the Attention Matrix \(A\) approximates the metric tensor \(g_{ij}\) locally.
2.2 The Field Equations¶
Can we derive an equivalent to Einstein's Field Equations? $$ G_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu} $$ Where \(T_{\mu\nu}\) represents the distribution of information in the training corpus.
3. Implications¶
3.1 Hallucination as Geodesic Deviation¶
Hallucination occurs when the trajectory of the prompt enters a region of "flat" or "undefined" curvature (sparse data), causing the geodesic to diverge unpredictably.
3.2 Creativity as Tunneling¶
Creativity may be modeled as a "wormhole" or "tunneling" event where the model jumps between two disparate geodesics that are topologically close in the high-dimensional manifold but semantically distant in linear space.
4. References & Further Reading¶
4.1 Phenomenology of LLMs¶
- Kaplan, J., et al. (2020). Scaling Laws for Neural Language Models. (The "Ideal Gas Law" of AI).
- Vaswani, A., et al. (2017). Attention Is All You Need. (The mechanics of the "Force").
4.2 Geometry of Information¶
- Amari, S. (2016). Information Geometry and Its Applications. (Riemannian geometry on probability distributions).
- Bronstein, M. M., et al. (2017). Geometric Deep Learning. (Generalizing DL to non-Euclidean domains).
4.3 Intrinsic Dimension¶
- Aghajanyan, A., et al. (2020). Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning.
- Li, C., et al. (2018). Measuring the Intrinsic Dimension of Objective Landscapes.
4.4 Category Theory¶
- Gavranović, B., et al. Categorical Deep Learning. (Compositionality as Functors).
5. Next Steps¶
- Formalize the definition of the "Semantic Metric Tensor."
- Test if Attention Weights obey the properties of a Riemannian Metric (symmetry, positive definiteness).
- Simulate "Geodesic Deviation" in a small Transformer to see if it predicts hallucination.