LieRE: Lie Rotational Positional Encodings

Abstract: Transformer architectures depend on explicit position encodings to capture token positional information. Rotary Position Encoding (RoPE) has emerged as a popular choice in LLMs due to its efficient encoding of relative position information through key-query rotations. However, RoPE faces significant limitations beyond language processing: it is constrained to one-dimensional sequence data and, even with learnable phases, offers limited representational capacity. We address these challenges with Lie Relative Encodings (LieRE), which generalizes RoPE to high-dimensional rotation matrices by leveraging their Lie group structure. Through extensive evaluation on three image datasets across 2D and 3D classification tasks, LieRE achieves 1.5% improvement over state-of-the-art baselines on 2D tasks and 1% on 3D tasks, while demonstrating superior generalization to higher resolutions. Our implementation is computationally efficient, with results reproducible on 4 A100 GPUs in 30 minutes on CIFAR100. Our code is available at https://github.com/StanfordMIMI/LieRE.