Möbius addition
An operation in hyperbolic space analogous to Euclidean addition. It is non-commutative and non-associative but satisfies the group axioms, translates points along geodesics, and automatically contracts step size near the boundary, making it suitable for encoding hierarchical structures.
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The Event2Vec library provides several non-commutative or geometrized alternative spaces: hyperbolic space (Möbius addition), the κ-stereographic model, spherical space (SO(n) rotations), and the Heisenberg group.
The Möbius addition formula is a ⊕_c b = ((1 + 2c⟨a,b⟩ + c‖b‖²)a + (1 − c‖a‖²)b) / (1 + 2c⟨a,b⟩ + c²‖a‖²‖b‖²), where c is the curvature; it is neither commutative nor associative, but satisfies the group axioms.
All Facts (2)
The Event2Vec library provides several non-commutative or geometrized alternative spaces: hyperbolic space (Möbius addition), the κ-stereographic model, spherical space (SO(n) rotations), and the Heisenberg group.
50%UnverifiedThe Möbius addition formula is a ⊕_c b = ((1 + 2c⟨a,b⟩ + c‖b‖²)a + (1 − c‖a‖²)b) / (1 + 2c⟨a,b⟩ + c²‖a‖²‖b‖²), where c is the curvature; it is neither commutative nor associative, but satisfies the group axioms.
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