Clifford Algebraic Rotor Embeddings : Maybe embeddings should start to CARE

Sameeksha Sriram, Timo Lüddecke, Ayush Paliwal, Alexander S. Ecker, Chase van de Geijn

Track: Extended Abstract Track

Keywords: Positional Encoding, RoPE, Geometric Algebra, Clifford Algebra, Rotors, Quaternions

TL;DR: We generalize Spherical Rotary Embeddings to Quaternion and Clifford Algebra based rotations

Abstract:

Rotary Positional Embeddings (RoPE) have demonstrated exceptional performance as a positional encoding method, consistently outperforming their baselines. While recent work has sought to extend RoPE to higher-dimensional inputs, many such extensions are non-commutative, thereby forfeiting RoPE’s shift-equivariance property. Spherical RoPE is one such non-commutative variant, motivated by the idea of rotating embedding vectors on spheres rather than circles. However, spherical rotations are inherently non-commutative, making the choice of rotation sequence ambiguous. In this work, we explore a quaternion-based approach—Quaternion Rotary Embeddings (QuatRo)—in place of Euler angles, leveraging quaternions’ ability to represent 3D rotations to parameterize the axes of rotation. We show Mixed RoPE and Spherical RoPE to be special cases of QuatRo.

Further, we propose a generalization of QuatRo to _Clifford Algebraic Rotary Embeddings_ (CARE) using geometric algebra. Viewing quaternions as the even subalgebra of $Cl(3,0,0)$, we extend the notion of rotary embeddings from quaternions to Clifford rotors acting on multivectors. This formulation enables two key generalizations: (1) extending rotary embeddings to arbitrary dimensions, and (2) encoding positional information in multivectors of multiple grades, not just vectors. We present preliminary experiments comparing spherical, quaternion, and Clifford-based rotary embeddings.

Submission Number: 123