Group Representational Position Encoding
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911682583855104 |
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| author | Zhang, Yifan Chen, Zixiang Liu, Yifeng Qin, Zhen Yuan, Huizhuo Xu, Kangping Yuan, Yang Gu, Quanquan Yao, Andrew Chi-Chih |
| author_facet | Zhang, Yifan Chen, Zixiang Liu, Yifeng Qin, Zhen Yuan, Huizhuo Xu, Kangping Yuan, Yang Gu, Quanquan Yao, Andrew Chi-Chih |
| contents | 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 $\operatorname{SO}(d)$ and (ii) additive logit biases (Additive GRAPE) arising from unipotent actions in the general linear group $\mathrm{GL}$. In Multiplicative GRAPE, a position $n \in \mathbb{Z}$ (or $t \in \mathbb{R}$) acts as $\mathbf{G}(n) = \exp(n \, ω\, \mathbf{L})$ with a rank-2 skew-symmetric generator $\mathbf{L} \in \mathbb{R}^{d \times d}$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_07805 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Group Representational Position Encoding Zhang, Yifan Chen, Zixiang Liu, Yifeng Qin, Zhen Yuan, Huizhuo Xu, Kangping Yuan, Yang Gu, Quanquan Yao, Andrew Chi-Chih Machine Learning Artificial Intelligence Computation and Language 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 $\operatorname{SO}(d)$ and (ii) additive logit biases (Additive GRAPE) arising from unipotent actions in the general linear group $\mathrm{GL}$. In Multiplicative GRAPE, a position $n \in \mathbb{Z}$ (or $t \in \mathbb{R}$) acts as $\mathbf{G}(n) = \exp(n \, ω\, \mathbf{L})$ with a rank-2 skew-symmetric generator $\mathbf{L} \in \mathbb{R}^{d \times d}$, 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. |
| title | Group Representational Position Encoding |
| topic | Machine Learning Artificial Intelligence Computation and Language |
| url | https://arxiv.org/abs/2512.07805 |