Polaris separates semantic meaning from hierarchical structure in embeddings via angular geometry and radius on a hypersphere, yielding up to 19-point gains in taxonomy expansion retrieval over baselines.
Finally, by Slutsky’s Theorem, the product of these converging terms is: ⟨z,u⟩= N√ d · 1q 1 d ∥x∥2 2 P − →0·1 1 = 0
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Polaris: Coupled Orbital Polar Embeddings for Hierarchical Concept Learning
Polaris separates semantic meaning from hierarchical structure in embeddings via angular geometry and radius on a hypersphere, yielding up to 19-point gains in taxonomy expansion retrieval over baselines.