# Group Representational Position Encoding

arXiv: https://arxiv.org/abs/2512.07805 · HTML: https://mlsanity.com/paper/2512.07805/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 50 references that are in the map.
- First posted on arXiv 8 Dec 2025.

## Builds on (most cited first, 30 of 45)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Longformer: The Long-Document Transformer](https://mlsanity.com/paper/2004.05150/) (2020)
- [DeBERTa: Decoding-enhanced BERT with Disentangled Attention](https://mlsanity.com/paper/2006.03654/) (2020)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context](https://mlsanity.com/paper/1901.02860/) (2019)
- [Big Bird: Transformers for Longer Sequences](https://mlsanity.com/paper/2007.14062/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Self-Attention with Relative Position Representations](https://mlsanity.com/paper/1803.02155/) (2018)
- [Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation](https://mlsanity.com/paper/2108.12409/) (2021)
- [Extending Context Window of Large Language Models via Positional Interpolation](https://mlsanity.com/paper/2306.15595/) (2023)
- [YaRN: Efficient Context Window Extension of Large Language Models](https://mlsanity.com/paper/2309.00071/) (2023)
- [The Impact of Positional Encoding on Length Generalization in Transformers](https://mlsanity.com/paper/2305.19466/) (2023)
- [Effective Long-Context Scaling of Foundation Models](https://mlsanity.com/paper/2309.16039/) (2023)
- [Rotary Position Embedding for Vision Transformer](https://mlsanity.com/paper/2403.13298/) (2024)
- [Rethinking Positional Encoding in Language Pre-training](https://mlsanity.com/paper/2006.15595/) (2020)
- [Transformer Language Models without Positional Encodings Still Learn Positional Information](https://mlsanity.com/paper/2203.16634/) (2022)
- [LLM Maybe LongLM: Self-Extend LLM Context Window Without Tuning](https://mlsanity.com/paper/2401.01325/) (2024)
- [A Length-Extrapolatable Transformer](https://mlsanity.com/paper/2212.10554/) (2022)
- [Randomized Positional Encodings Boost Length Generalization of Transformers](https://mlsanity.com/paper/2305.16843/) (2023)
- [Round and Round We Go! What makes Rotary Positional Encodings useful?](https://mlsanity.com/paper/2410.06205/) (2024)
- [PoSE: Efficient Context Window Extension of LLMs via Positional Skip-wise Training](https://mlsanity.com/paper/2309.10400/) (2023)
- [KERPLE: Kernelized Relative Positional Embedding for Length Extrapolation](https://mlsanity.com/paper/2205.09921/) (2022)
- [Learning to Encode Position for Transformer with Continuous Dynamical Model](https://mlsanity.com/paper/2003.09229/) (2020)
- [Contextual Position Encoding: Learning to Count What's Important](https://mlsanity.com/paper/2405.18719/) (2024)

## Built on by (most cited first, 5 of 5)

- [Jordan-RoPE: Non-Semisimple Relative Positional Encoding via Complex Jordan Blocks](https://mlsanity.com/paper/2605.04217/) (2026)
- [PJ-RoPE: A Fourier-Jet-Affine Position Space for Relative Attention](https://mlsanity.com/paper/2606.05345/) (2026)
- [The Token Is a Group Element: On Lie-Algebra Attention over Matrix Lie Groups](https://mlsanity.com/paper/2606.20547/) (2026)
- [MinkowskiPE: Minkowski Positional Encoding for Spatiotemporal Perception](https://mlsanity.com/paper/2609.33804/) (2026)
- [How Local Mixing Encodes Relative Position in Global NoPE Attention](https://mlsanity.com/paper/2609.38109/) (2026)

## Abstract (arXiv)

We present GRAPE (Group Representational Position Encoding), a unified framework for positional encoding based on group actions. GRAPE unifies two families of mechanisms: (i) multiplicative rotations (Multiplicative GRAPE) in operatornameSO(d) and (ii) additive logit biases (Additive GRAPE) arising from unipotent actions in the general linear group GL. In Multiplicative GRAPE, a position n ∈ mathbbZ (or t ∈ ℝ) acts as G(n) = exp(n , ω , L) with a rank-2 skew-symmetric generator L ∈ ℝd × d, yielding a relative, compositional, norm-preserving map with a closed-form matrix exponential. RoPE is recovered exactly when the d/2 planes correspond to canonical coordinate pairs with a log-uniform spectrum. Learned commuting subspaces and compact non-commuting mixtures strictly extend this geometry to capture cross-subspace feature coupling at O(d) and O(r d) cost per head, respectively. In Additive GRAPE, additive logits arise from rank-1 (or low-rank) unipotent actions, recovering ALiBi and the Forgetting Transformer (FoX) as exact special cases while preserving an exact relative law and streaming cacheability. Overall, GRAPE provides a principled design space for positional geometry in long-context models, subsuming RoPE and ALiBi as special cases. Project page: https://github.com/model-architectures/GRAPE.