Clifford Algebraic Rotor Embeddings : Maybe embeddings should start to CARE

TL;DR

This paper introduces Clifford Algebraic Rotary Embeddings (CARE), a novel generalization of rotary positional embeddings using geometric algebra, enabling higher-dimensional and multivector-based positional encoding with promising preliminary results.

Contribution

The paper proposes CARE, a new framework that extends rotary embeddings to arbitrary dimensions and multivectors via Clifford algebra, unifying and generalizing previous approaches.

Findings

QuatRo encompasses spherical RoPE as a special case

CARE enables encoding in higher dimensions and multiple grades

Preliminary experiments compare spherical, quaternion, and Clifford embeddings

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.