[论文] The Token Is a Group Element: On Lie-Algebra Attention over Matrix Lie...

论文概要

研究领域: CV
作者: Przemyslaw Musialski
发布时间: 2025-06-20
arXiv: 2506.16802

中文摘要

本文置注意力令牌于群上：一令牌乃矩阵李群G之一元素\(g_i\)——一纯变换，无特征负载，亦无外在动作携之。据作者所知，此为首种其令牌为赤裸矩阵李群元素之注意力构造：其得分乃相对位姿之闭式代数范数，而非习得核，且能达每一种基于不可约表示或满射指数方法所必排斥之仿射全帧群。此构造谓之李代数注意力。

令牌既为群元素，其余皆无需惯常表示论之机括而自成。相对几何为一对之典范，即\(g_i^{-1} g_j\)，故成对不变量\(w_{ij}=log(g_i^{-1} g_j)\)乃内在而非设计；对角G作用下之等变性为自明，上循环条件亦自动成立。注意力得分乃负平方代数范数：分块加权Frobenius内积下之典范邻近核，无需不可约表示、球谐、Clebsch-Gordan乘积或习得核。此构造适用于任一矩阵李群，于所选对数图表含相对位姿者，包括具尺度与剪切之非紧非阿贝尔仿射群——此乃任何向量令牌注意力方法所不及。

三项序列完形实验，于SE(2)、SO(3)与Aff(2)上证实之：闭式得分与同一不变量上之习得MLP核相埒，且在SE(2)上优之，用参数少五十至八十倍；而向量令牌基线则以五至十二个数量级破坏等变性。

原文摘要

We place the attention token on the group: a token is an element \(g_i\) of a matrix Lie group \(G\) -- a bare transformation, with no feature payload and no external action \(\rho(g)\) carrying it. To our knowledge this is the first attention construction whose tokens are bare matrix Lie group elements: their score is the closed-form algebra norm of the relative pose rather than a learned kernel, and it reaches the affine full-frame groups that every irrep- or surjective-exp-based method must exclude. We call it Lie-Algebra Attention. Once tokens are group elements, the rest follows with none of the usual representation-theoretic machinery. The relative geometry of a pair is canonical, \(g_i^{-1} g_j\), so the pairwise invariant \(w_{ij} = \log(g_i^{-1} g_j)\) is intrinsic rather than designed; equ...

自动采集于 2026-06-20

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