Artificial Intelligence · 12.08.2026, 04:55 UTC
Riemann GeoResolver: A Non-Euclidean Attention Framework from Euclidean Resolver to Hyperbolic-Spherical Geometry
| Schweregrad | info |
|---|---|
| Kategorie | Artificial Intelligence |
| Quelle | arXiv cs.CL ↗ |
| Veröffentlicht | 12.08.2026 UTC |
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arXiv:2608.10416v1 Announce Type: cross Abstract: We present a theoretical foundation for inverse-distance attention, from its Euclidean prototype (Resolver) to its non-Euclidean realization (Riemann GeoResolver). The Euclidean part establishes three core theorems: (1) circuit separation---IDA achieves exact retrieval with $\mathcal{O}(1)$ resources while softmax requires $\Omega((\log n)^2)$ width; (2) a Polyak--Lojasiewicz inequality with $\Omega(e^{\Delta^2/\sqrt{d}}/\Delta^2)$ stronger constant than softmax, implying linear convergence, $\mathcal{O}(\log n)$ Lipschitz scaling under a low-rank/clustering assumption, $\Theta(1)$ Hessian spread, and absence of spurious local minima; (3) a width-independent effective rank bound that limits noise memorization---softmax memorizes arbitrary labels when $d_h\ge n$, while IDA limits test error to $\mathcal{O}(\eta^2)$. The non-Euclidean extension then builds upon this prototype, replacing Euclidean distance with hyperbolic geodesic distance for storage and spherical geodesic distance for routing. The Riemann GeoResolver framework comprises ten integrated modules: four HIDA operators spanning $\Theta(n^2)$ to $\Theta(1)$ per token; Hyperbolic Curvature Compression (HCC) with provable error bounds; HyperGate with gradient lower-bound theorem; Spherical Inverse Distance Attention (SIDA) with sphere-analog PL inequalities; Dynamic Memory Genesis (DMG) with $\mathcal{O}(\log T)$ regret bounds; and Geodesic Sparse Routing (GSR) with quality and communication bounds. The Euclidean theorems are proved in full; the non-Euclidean …